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Improve Katex error handling and fix handling of large expressions
* Make throwOnError=true the new default
* Handle JS errors as part of the RPC request/response flow
* Return a new Result type with .Err on it
This enables constructs on the form:
```handlebars
{{ with transform.ToMath "c = \\foo{a^2 + b^2}" }}
{{ with .Err }}
{{ warnf "error: %s" . }}
{{ else }}
{{ . }}
{{ end }}
{{ end }}
```
Note that the new `Result` type behaves like `template.HTML` (or a string if needed) when printed, but it will panic if in a error state.
Closes #12748
This commit is contained in:
Bjørn Erik Pedersen
parent
e42263529c
commit
e1e1baa1bd
14 changed files with 569 additions and 24 deletions
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@ -120,3 +120,27 @@ var InvocationCounter atomic.Int64
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func NewBool(b bool) *bool {
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return &b
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}
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// PrintableValueProvider is implemented by types that can provide a printable value.
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type PrintableValueProvider interface {
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PrintableValue() any
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}
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var _ PrintableValueProvider = Result[any]{}
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// Result is a generic result type.
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type Result[T any] struct {
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// The result value.
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Value T
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// The error value.
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Err error
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}
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// PrintableValue returns the value or panics if there is an error.
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func (r Result[T]) PrintableValue() any {
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if r.Err != nil {
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panic(r.Err)
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}
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return r.Value
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}
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@ -30,13 +30,23 @@ export function readInput(handle) {
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currentLine = [...currentLine, ...buffer.subarray(0, bytesRead)];
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// Check for newline. If not, we need to read more data.
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if (!currentLine.includes(10)) {
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continue;
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}
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// Split array into chunks by newline.
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let i = 0;
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for (let j = 0; i < currentLine.length; i++) {
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if (currentLine[i] === 10) {
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const chunk = currentLine.splice(j, i + 1);
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const arr = new Uint8Array(chunk);
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const json = JSON.parse(new TextDecoder().decode(arr));
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let json;
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try {
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json = JSON.parse(new TextDecoder().decode(arr));
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} catch (e) {
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throw new Error(`Error parsing JSON '${new TextDecoder().decode(arr)}' from stdin: ${e.message}`);
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}
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handle(json);
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j = i + 1;
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}
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@ -1,2 +1,2 @@
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(()=>{function i(r){let e=[],a=new Uint8Array(1024);for(;;){let n=0;try{n=Javy.IO.readSync(0,a)}catch(o){if(o.message.includes("os error 29"))break;throw new Error("Error reading from stdin")}if(n<0)throw new Error("Error reading from stdin");if(n===0)break;e=[...e,...a.subarray(0,n)];let t=0;for(let o=0;t<e.length;t++)if(e[t]===10){let f=e.splice(o,t+1),s=new Uint8Array(f),u=JSON.parse(new TextDecoder().decode(s));r(u),o=t+1}e=e.slice(t)}}function d(r){let c=new TextEncoder().encode(JSON.stringify(r)+`
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`),e=new Uint8Array(c);Javy.IO.writeSync(1,e)}var l=function(r){d({header:r.header,data:{greeting:"Hello "+r.data.name+"!"}})};console.log("Greet module loaded");i(l);})();
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(()=>{function s(r){let e=[],c=new Uint8Array(1024);for(;;){let n=0;try{n=Javy.IO.readSync(0,c)}catch(o){if(o.message.includes("os error 29"))break;throw new Error("Error reading from stdin")}if(n<0)throw new Error("Error reading from stdin");if(n===0)break;if(e=[...e,...c.subarray(0,n)],!e.includes(10))continue;let t=0;for(let o=0;t<e.length;t++)if(e[t]===10){let u=e.splice(o,t+1),d=new Uint8Array(u),a;try{a=JSON.parse(new TextDecoder().decode(d))}catch(l){throw new Error(`Error parsing JSON '${new TextDecoder().decode(d)}' from stdin: ${l.message}`)}r(a),o=t+1}e=e.slice(t)}}function f(r){let i=new TextEncoder().encode(JSON.stringify(r)+`
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`),e=new Uint8Array(i);Javy.IO.writeSync(1,e)}var w=function(r){f({header:r.header,data:{greeting:"Hello "+r.data.name+"!"}})};console.log("Greet module loaded");s(w);})();
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File diff suppressed because one or more lines are too long
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@ -5,7 +5,14 @@ const render = function (input) {
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const data = input.data;
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const expression = data.expression;
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const options = data.options;
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writeOutput({ header: input.header, data: { output: katex.renderToString(expression, options) } });
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const header = input.header;
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try {
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const output = katex.renderToString(expression, options);
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writeOutput({ header: header, data: { output: output } });
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} catch (e) {
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header.err = e.message;
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writeOutput({ header: header });
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}
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};
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readInput(render);
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@ -39,7 +39,6 @@ type KatexOptions struct {
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MinRuleThickness float64 `json:"minRuleThickness"`
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// If true, KaTeX will throw a ParseError when it encounters an unsupported command.
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// For internal use only, for now.
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ThrowOnError bool `json:"throwOnError"`
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}
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@ -31,6 +31,9 @@ var quickjsWasm []byte
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type Header struct {
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Version string `json:"version"`
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ID uint32 `json:"id"`
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// Set in the response if there was an error.
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Err string `json:"err"`
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}
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type Message[T any] struct {
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@ -179,7 +182,7 @@ func (d *dispatcher[Q, R]) input() {
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for d.inOut.dec.More() {
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var r Message[R]
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if err := d.inOut.dec.Decode(&r); err != nil {
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inputErr = err
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inputErr = fmt.Errorf("decoding response: %w", err)
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break
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}
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Binary file not shown.
Binary file not shown.
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@ -29,7 +29,8 @@ const (
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// indirect returns the value, after dereferencing as many times
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// as necessary to reach the base type (or nil).
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func indirect(a any) any {
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// Signature modified by Hugo. TODO(bep) script this.
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func doIndirect(a any) any {
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if a == nil {
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return nil
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}
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@ -14,6 +14,7 @@
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package template
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import (
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"github.com/gohugoio/hugo/common/types"
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template "github.com/gohugoio/hugo/tpl/internal/go_templates/texttemplate"
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)
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@ -39,3 +40,14 @@ func (t *Template) Prepare() (*template.Template, error) {
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func StripTags(html string) string {
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return stripTags(html)
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}
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func indirect(a any) any {
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in := doIndirect(a)
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// We have a special Result type that we want to unwrap when printed.
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if pp, ok := in.(types.PrintableValueProvider); ok {
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return pp.PrintableValue()
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}
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return in
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}
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373
tpl/transform/testdata/large-katex.md
vendored
Normal file
373
tpl/transform/testdata/large-katex.md
vendored
Normal file
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@ -0,0 +1,373 @@
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---
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title: Large KaTeX
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source: https://math.stackexchange.com/questions/8337/different-methods-to-compute-sum-limits-k-1-infty-frac1k2-basel-pro
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license: https://creativecommons.org/licenses/by-sa/4.0/
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---
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As I have heard people did not trust Euler when he first discovered the formula (solution of the Basel problem) $$\zeta(2)=\sum_{k=1}^\infty \frac{1}{k^2}=\frac{\pi^2}{6}.$$ However, Euler was Euler and he gave other proofs.
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I believe many of you know some nice proofs of this, can you please share it with us?
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Hans Lundmark
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OK, here's my favorite. I thought of this after reading a proof from the book "Proofs from the book" by Aigner & Ziegler, but later I found more or less the same proof as mine in a paper published a few years earlier by Josef Hofbauer. On Robin's list, the proof most similar to this is number 9 (EDIT: ...which is actually the proof that I read in Aigner & Ziegler).
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When $0 < x < \pi/2$ we have $0<\sin x < x < \tan x$ and thus $$\frac{1}{\tan^2 x} < \frac{1}{x^2} < \frac{1}{\sin^2 x}.$$ Note that $1/\tan^2 x = 1/\sin^2 x - 1$. Split the interval $(0,\pi/2)$ into $2^n$ equal parts, and sum the inequality over the (inner) "gridpoints" $x_k=(\pi/2) \cdot (k/2^n)$: $$\sum_{k=1}^{2^n-1} \frac{1}{\sin^2 x_k} - \sum_{k=1}^{2^n-1} 1 < \sum_{k=1}^{2^n-1} \frac{1}{x_k^2} < \sum_{k=1}^{2^n-1} \frac{1}{\sin^2 x_k}.$$ Denoting the sum on the right-hand side by $S_n$, we can write this as $$S_n - (2^n - 1) < \sum_{k=1}^{2^n-1} \left( \frac{2 \cdot 2^n}{\pi} \right)^2 \frac{1}{k^2} < S_n.$$
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Although $S_n$ looks like a complicated sum, it can actually be computed fairly easily. To begin with, $$\frac{1}{\sin^2 x} + \frac{1}{\sin^2 (\frac{\pi}{2}-x)} = \frac{\cos^2 x + \sin^2 x}{\cos^2 x \cdot \sin^2 x} = \frac{4}{\sin^2 2x}.$$ Therefore, if we pair up the terms in the sum $S_n$ except the midpoint $\pi/4$ (take the point $x_k$ in the left half of the interval $(0,\pi/2)$ together with the point $\pi/2-x_k$ in the right half) we get 4 times a sum of the same form, but taking twice as big steps so that we only sum over every other gridpoint; that is, over those gridpoints that correspond to splitting the interval into $2^{n-1}$ parts. And the midpoint $\pi/4$ contributes with $1/\sin^2(\pi/4)=2$ to the sum. In short, $$S_n = 4 S_{n-1} + 2.$$ Since $S_1=2$, the solution of this recurrence is $$S_n = \frac{2(4^n-1)}{3}.$$ (For example like this: the particular (constant) solution $(S_p)_n = -2/3$ plus the general solution to the homogeneous equation $(S_h)_n = A \cdot 4^n$, with the constant $A$ determined by the initial condition $S_1=(S_p)_1+(S_h)_1=2$.)
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We now have $$ \frac{2(4^n-1)}{3} - (2^n-1) \leq \frac{4^{n+1}}{\pi^2} \sum_{k=1}^{2^n-1} \frac{1}{k^2} \leq \frac{2(4^n-1)}{3}.$$ Multiply by $\pi^2/4^{n+1}$ and let $n\to\infty$. This squeezes the partial sums between two sequences both tending to $\pi^2/6$. Voilà!
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Américo Tavares
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We can use the function $f(x)=x^{2}$ with $-\pi \leq x\leq \pi $ and find its expansion into a trigonometric Fourier series
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$$\dfrac{a_{0}}{2}+\sum_{n=1}^{\infty }(a_{n}\cos nx+b_{n}\sin nx),$$
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which is periodic and converges to $f(x)$ in $[-\pi, \pi] $.
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Observing that $f(x)$ is even, it is enough to determine the coefficients
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$$a_{n}=\dfrac{1}{\pi }\int_{-\pi }^{\pi }f(x)\cos nx\;dx\qquad n=0,1,2,3,...,$$
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because
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$$b_{n}=\dfrac{1}{\pi }\int_{-\pi }^{\pi }f(x)\sin nx\;dx=0\qquad n=1,2,3,... .$$
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For $n=0$ we have
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$$a_{0}=\dfrac{1}{\pi }\int_{-\pi }^{\pi }x^{2}dx=\dfrac{2}{\pi }\int_{0}^{\pi }x^{2}dx=\dfrac{2\pi ^{2}}{3}.$$
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And for $n=1,2,3,...$ we get
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$$a_{n}=\dfrac{1}{\pi }\int_{-\pi }^{\pi }x^{2}\cos nx\;dx$$
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$$=\dfrac{2}{\pi }\int_{0}^{\pi }x^{2}\cos nx\;dx=\dfrac{2}{\pi }\times \dfrac{ 2\pi }{n^{2}}(-1)^{n}=(-1)^{n}\dfrac{4}{n^{2}},$$
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because
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$$\int x^2\cos nx\;dx=\dfrac{2x}{n^{2}}\cos nx+\left( \frac{x^{2}}{ n}-\dfrac{2}{n^{3}}\right) \sin nx.$$
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Thus
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$$f(x)=\dfrac{\pi ^{2}}{3}+\sum_{n=1}^{\infty }\left( (-1)^{n}\dfrac{4}{n^{2}} \cos nx\right) .$$
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Since $f(\pi )=\pi ^{2}$, we obtain
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$$\pi ^{2}=\dfrac{\pi ^{2}}{3}+\sum_{n=1}^{\infty }\left( (-1)^{n}\dfrac{4}{ n^{2}}\cos \left( n\pi \right) \right) $$
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$$\pi ^{2}=\dfrac{\pi ^{2}}{3}+4\sum_{n=1}^{\infty }\left( (-1)^{n}(-1)^{n} \dfrac{1}{n^{2}}\right) $$
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$$\pi ^{2}=\dfrac{\pi ^{2}}{3}+4\sum_{n=1}^{\infty }\dfrac{1}{n^{2}}.$$
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Therefore
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$$\sum_{n=1}^{\infty }\dfrac{1}{n^{2}}=\dfrac{\pi ^{2}}{4}-\dfrac{\pi ^{2}}{12}= \dfrac{\pi ^{2}}{6}$$
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Second method (available on-line a few years ago) by Eric Rowland. From
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$$\log (1-t)=-\sum_{n=1}^{\infty}\dfrac{t^n}{n}$$
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and making the substitution $t=e^{ix}$ one gets the series expansion
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$$w=\text{Log}(1-e^{ix})=-\sum_{n=1}^{\infty }\dfrac{e^{inx}}{n}=-\sum_{n=1}^{ \infty }\dfrac{1}{n}\cos nx-i\sum_{n=1}^{\infty }\dfrac{1}{n}\sin nx,$$
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whose radius of convergence is $1$. Now if we take the imaginary part of both sides, the RHS becomes
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$$\Im w=-\sum_{n=1}^{\infty }\dfrac{1}{n}\sin nx,$$
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and the LHS
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$$\Im w=\arg \left( 1-\cos x-i\sin x\right) =\arctan \dfrac{-\sin x}{ 1-\cos x}.$$
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Since
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$$\arctan \dfrac{-\sin x}{1-\cos x}=-\arctan \dfrac{2\sin \dfrac{x}{2}\cdot \cos \dfrac{x}{2}}{2\sin ^{2}\dfrac{x}{2}}$$
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$$=-\arctan \cot \dfrac{x}{2}=-\arctan \tan \left( \dfrac{\pi }{2}-\dfrac{x}{2} \right) =\dfrac{x}{2}-\dfrac{\pi }{2},$$
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the following expansion holds
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$$\dfrac{\pi }{2}-\frac{x}{2}=\sum_{n=1}^{\infty }\dfrac{1}{n}\sin nx.\qquad (\ast )$$
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Integrating the identity $(\ast )$, we obtain
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$$\dfrac{\pi }{2}x-\dfrac{x^{2}}{4}+C=-\sum_{n=1}^{\infty }\dfrac{1}{n^{2}}\cos nx.\qquad (\ast \ast )$$
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Setting $x=0$, we get the relation between $C$ and $\zeta (2)$
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$$C=-\sum_{n=1}^{\infty }\dfrac{1}{n^{2}}=-\zeta (2).$$
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And for $x=\pi $, since
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$$\zeta (2)=2\sum_{n=1}^{\infty }\dfrac{(-1)^{n-1}}{n^{2}},$$
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we deduce
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$$\dfrac{\pi ^{2}}{4}+C=-\sum_{n=1}^{\infty }\dfrac{1}{n^{2}}\cos n\pi =\sum_{n=1}^{\infty }\dfrac{(-1)^{n-1}}{n^{2}}=\dfrac{1}{2}\zeta (2)=-\dfrac{1}{ 2}C.$$
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Solving for $C$
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$$C=-\dfrac{\pi ^{2}}{6},$$
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we thus prove
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$$\zeta (2)=\dfrac{\pi ^{2}}{6}.$$
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Note: this 2nd method can generate all the zeta values $\zeta (2n)$ by integrating repeatedly $(\ast\ast )$. This is the reason why I appreciate it. Unfortunately it does not work for $\zeta (2n+1)$.
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Note also the $$C=-\dfrac{\pi ^{2}}{6}$$ can be obtained by integrating $(\ast\ast )$ and substitute $$x=0,x=\pi$$ respectively.
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AD.
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Here is an other one which is more or less what Euler did in one of his proofs.
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The function $\sin x$ where $x\in\mathbb{R}$ is zero exactly at $x=n\pi$ for each integer $n$. If we factorized it as an infinite product we get
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$$\sin x = \cdots\left(1+\frac{x}{3\pi}\right)\left(1+\frac{x}{2\pi}\right)\left(1+\frac{x}{\pi}\right)x\left(1-\frac{x}{\pi}\right)\left(1-\frac{x}{2\pi}\right)\left(1-\frac{x}{3\pi}\right)\cdots =$$ $$= x\left(1-\frac{x^2}{\pi^2}\right)\left(1-\frac{x^2}{2^2\pi^2}\right)\left(1-\frac{x^2}{3^2\pi^2}\right)\cdots\quad.$$
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We can also represent $\sin x$ as a Taylor series at $x=0$:
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$$\sin x = x - \frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\cdots\quad.$$
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Multiplying the product and identifying the coefficient of $x^3$ we see that
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$$\frac{x^3}{3!}=x\left(\frac{x^2}{\pi^2} + \frac{x^2}{2^2\pi^2}+ \frac{x^2}{3^2\pi^2}+\cdots\right)=x^3\sum_{n=1}^{\infty}\frac{1}{n^2\pi^2}$$ or $$\sum_{n=1}^\infty\frac{1}{n^2}=\frac{\pi^2}{6}.$$
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Alfredo Z.
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Define the following series for $ x > 0 $
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$$\frac{\sin x}{x} = 1 - \frac{x^2}{3!}+\frac{x^4}{5!}-\frac{x^6}{7!}+\cdots\quad.$$
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Now substitute $ x = \sqrt{y}\ $ to arrive at
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$$\frac{\sin \sqrt{y}\ }{\sqrt{y}\ } = 1 - \frac{y}{3!}+\frac{y^2}{5!}-\frac{y^3}{7!}+\cdots\quad.$$
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if we find the roots of $\frac{\sin \sqrt{y}\ }{\sqrt{y}\ } = 0 $ we find that
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$ y = n^2\pi^2\ $ for $ n \neq 0 $ and $ n $ in the integers
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With all of this in mind, recall that for a polynomial
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$ P(x) = a_{n}x^n + a_{n-1}x^{n-1} +\cdots+a_{1}x + a_{0} $ with roots $ r_{1}, r_{2}, \cdots , r_{n} $
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$$\frac{1}{r_{1}} + \frac{1}{r_{2}} + \cdots + \frac{1}{r_{n}} = -\frac{a_{1}}{a_{0}}$$
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Treating the above series for $ \frac{\sin \sqrt{y}\ }{\sqrt{y}\ } $ as polynomial we see that
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$$\frac{1}{1^2\pi^2} + \frac{1}{2^2\pi^2} + \frac{1}{3^2\pi^2} + \cdots = -\frac{-\frac{1}{3!}}{1}$$
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then multiplying both sides by $ \pi^2 $ gives the desired series.
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$$\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \cdots = \frac{\pi^2}{6}$$
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Nameless
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This method apparently was used by Tom Apostol in $1983$. I will outline the main ideas of the proof, the details can be found in here or this presentation (page $27$)
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Consider
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$$\begin{align} \int_{0}^{1} \int_{0}^{1} \frac{1}{1 - xy} dy dx &= \int_{0}^{1} \int_{0}^{1} \sum_{n \geq 0} (xy)^n dy dx \\ &= \sum_{n \geq 0} \int_{0}^{1} \int_{0}^{1} x^n y^n dy dx \\ &= \sum_{n \geq 1} \frac{1}{n^2} \\ \end{align}$$
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You can verify that the left hand side is indeed $\frac{\pi^2}{6}$ by letting $x = u - v$ and $y = v + u.$
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Qiaochu Yuan
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I have two favorite proofs. One is the last proof in Robin Chapman's collection; you really should take a look at it.
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The other is a proof that generalizes to the evaluation of $\zeta(2n)$ for all $n$, although I'll do it "Euler-style" to shorten the presentation. The basic idea is that meromorphic functions have infinite partial fraction decompositions that generalize the partial fraction decompositions of rational functions.
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The particular function we're interested in is $B(x) = \frac{x}{e^x - 1}$, the exponential generating function of the Bernoulli numbers $B_n$. $B$ is meromorphic with poles at $x = 2 \pi i n, n \in \mathbb{Z}$, and at these poles it has residue $2\pi i n$. It follows that we can write, a la Euler,
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$$\frac{x}{e^x - 1} = \sum_{n \in \mathbb{Z}} \frac{2\pi i n}{x - 2 \pi i n} = \sum_{n \in \mathbb{Z}} - \left( \frac{1}{1 - \frac{x}{2\pi i n}} \right).$$
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Now we can expand each of the terms on the RHS as a geometric series, again a la Euler, to obtain
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$$\frac{x}{e^x - 1} = - \sum_{n \in \mathbb{Z}} \sum_{k \ge 0} \left( \frac{x}{2\pi i n} \right)^k = \sum_{k \ge 0} (-1)^{n+1} \frac{2 \zeta(2n)}{(2\pi )^{2n}} x^{2n}$$
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because, after rearranging terms, the sum over odd powers cancels out and the sum over even powers doesn't. (This is one indication of why there is no known closed form for $\zeta(2n+1)$.) Equating terms on both sides, it follows that
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$$B_{2n} = (-1)^{n+1} \frac{2 \zeta(2n)}{(2\pi)^{2n}}$$
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or
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$$\zeta(2n) = (-1)^{n+1} \frac{B_{2n} (2\pi)^{2n}}{2}$$
|
||||
as desired. To compute $\zeta(2)$ it suffices to compute that $B_2 = \frac{1}{6}$, which then gives the usual answer.
|
||||
Here is one more nice proof, I learned it from Grisha Mikhalkin:
|
||||
Lemma: Let $Z$ be a complex curve in $\mathbb{C}^2$. Let $R(Z) \subset \mathbb{R}^2$ be the projection of $Z$ onto its real parts and $I(Z)$ the projection onto its complex parts. If these projections are both one to one, then the area of $R(Z)$ is equal to the area of $I(Z)$.
|
||||
Proof: There is an obvious map from $R(Z)$ to $I(Z)$, given by lifting $(x_1, x_2) \in R(Z)$ to $(x_1+i y_1, x_2 + i y_2) \in Z$, and then projecting to $(y_1, y_2) \in I(Z)$. We must prove this map has Jacobian $1$. WLOG, translate $(x_1, y_1, x_2, y_2)$ to $(0,0,0,0)$ and let $Z$ obey $\partial z_2/\partial z_1 = a+bi$ near $(0,0)$. To first order, we have $x_2 = a x_1 - b y_1$ and $y_2 = a y_1 + b x_1$. So $y_1 = (a/b) x_1 - (1/b) x_2$ and $y_2 = (a^2 + b^2)/b x_1 - (a/b) x_2$. So the derivative of $(x_1, x_2) \mapsto (y_1, y_2)$ is $\left( \begin{smallmatrix} a/b & - 1/b \\ (a^2 + b^2)/b & -a/b \end{smallmatrix} \right)$ and the Jacobian is $1$. QED
|
||||
Now, consider the curve $e^{-z_1} + e^{-z_2} = 1$, where $z_1$ and $z_2$ obey the following inequalities: $x_1 \geq 0$, $x_2 \geq 0$, $-\pi \leq y_1 \leq 0$ and $0 \leq y_2 \leq \pi$.
|
||||
Given a point on $e^{-z_1} + e^{-z_2} = 1$, consider the triangle with vertices at $0$, $e^{-z_1}$ and $e^{-z_1} + e^{-z_2} = 1$. The inequalities on the $y$'s states that the triangle should lie above the real axis; the inequalities on the $x$'s state the horizontal base should be the longest side.
|
||||
Projecting onto the $x$ coordinates, we see that the triangle exists if and only if the triangle inequality $e^{-x_1} + e^{-x_2} \geq 1$ is obeyed. So $R(Z)$ is the region under the curve $x_2 = - \log(1-e^{-x_1})$. The area under this curve is $$\int_{0}^{\infty} - \log(1-e^{-x}) dx = \int_{0}^{\infty} \sum \frac{e^{-kx}}{k} dx = \sum \frac{1}{k^2}.$$
|
||||
Now, project onto the $y$ coordinates. Set $(y_1, y_2) = (-\theta_1, \theta_2)$ for convenience, so the angles of the triangle are $(\theta_1, \theta_2, \pi - \theta_1 - \theta_2)$. The largest angle of a triangle is opposite the largest side, so we want $\theta_1$, $\theta_2 \leq \pi - \theta_1 - \theta_2$, plus the obvious inequalities $\theta_1$, $\theta_2 \geq 0$. So $I(Z)$ is the quadrilateral with vertices at $(0,0)$, $(0, \pi/2)$, $(\pi/3, \pi/3)$ and $(\pi/2, 0)$ and, by elementary geometry, this has area $\pi^2/6$.
|
||||
|
||||
David Speyer
|
||||
I'll post the one I know since it is Euler's, and is quite easy and stays in $\mathbb{R}$. (I'm guessing Euler didn't have tools like residues back then).
|
||||
|
||||
Peter Tamaroff
|
||||
Let
|
||||
$$s = {\sin ^{ - 1}}x$$
|
||||
Then
|
||||
$$\int\limits_0^{\frac{\pi }{2}} {sds} = \frac{{{\pi ^2}}}{8}$$
|
||||
But then
|
||||
$$\int\limits_0^1 {\frac{{{{\sin }^{ - 1}}x}}{{\sqrt {1 - {x^2}} }}dx} = \frac{{{\pi ^2}}}{8}$$
|
||||
Since
|
||||
$${\sin ^{ - 1}}x = \int {\frac{{dx}}{{\sqrt {1 - {x^2}} }}} = x + \frac{1}{2}\frac{{{x^3}}}{3} + \frac{{1 \cdot 3}}{{2 \cdot 4}}\frac{{{x^5}}}{5} + \frac{{1 \cdot 3 \cdot 5}}{{2 \cdot 4 \cdot 6}}\frac{{{x^7}}}{7} + \cdots $$
|
||||
We have
|
||||
$$\int\limits_0^1 {\left\{ {\frac{{dx}}{{\sqrt {1 - {x^2}} }}\int {\frac{{dx}}{{\sqrt {1 - {x^2}} }}} } \right\}} = \int\limits_0^1 {\left\{ {x + \frac{1}{2}\frac{{{x^3}}}{3}\frac{{dx}}{{\sqrt {1 - {x^2}} }} + \frac{{1 \cdot 3}}{{2 \cdot 4}}\frac{{{x^5}}}{5}\frac{{dx}}{{\sqrt {1 - {x^2}} }} + \cdots } \right\}} $$
|
||||
But
|
||||
$$\int\limits_0^1 {\frac{{{x^{2n + 1}}}}{{\sqrt {1 - {x^2}} }}dx} = \frac{{2n}}{{2n + 1}}\int\limits_0^1 {\frac{{{x^{2n - 1}}}}{{\sqrt {1 - {x^2}} }}dx} $$
|
||||
which yields
|
||||
$$\int\limits_0^1 {\frac{{{x^{2n + 1}}}}{{\sqrt {1 - {x^2}} }}dx} = \frac{{\left( {2n} \right)!!}}{{\left( {2n + 1} \right)!!}}$$
|
||||
since all powers are odd.
|
||||
This ultimately produces:
|
||||
$$\frac{{{\pi ^2}}}{8} = 1 + \frac{1}{2}\frac{1}{3}\left( {\frac{2}{3}} \right) + \frac{{1 \cdot 3}}{{2 \cdot 4}}\frac{1}{5}\left( {\frac{{2 \cdot 4}}{{3 \cdot 5}}} \right) + \frac{{1 \cdot 3 \cdot 5}}{{2 \cdot 4 \cdot 6}}\frac{1}{7}\left( {\frac{{2 \cdot 4 \cdot 6}}{{3 \cdot 5 \cdot 7}}} \right) \cdots $$
|
||||
$$\frac{{{\pi ^2}}}{8} = 1 + \frac{1}{{{3^2}}} + \frac{1}{{{5^2}}} + \frac{1}{{{7^2}}} + \cdots $$
|
||||
Let
|
||||
$$1 + \frac{1}{{{2^2}}} + \frac{1}{{{3^2}}} + \frac{1}{{{4^2}}} + \cdots = \omega $$
|
||||
Then
|
||||
$$\frac{1}{{{2^2}}} + \frac{1}{{{4^2}}} + \frac{1}{{{6^2}}} + \frac{1}{{{8^2}}} + \cdots = \frac{\omega }{4}$$
|
||||
Which means
|
||||
$$\frac{\omega }{4} + \frac{{{\pi ^2}}}{8} = \omega $$
|
||||
or
|
||||
$$\omega = \frac{{{\pi ^2}}}{6}$$
|
||||
|
||||
Mike Spivey
|
||||
The most recent issue of The American Mathematical Monthly (August-September 2011, pp. 641-643) has a new proof by Luigi Pace based on elementary probability. Here's the argument.
|
||||
Let $X_1$ and $X_2$ be independent, identically distributed standard half-Cauchy random variables. Thus their common pdf is $p(x) = \frac{2}{\pi (1+x^2)}$ for $x > 0$.
|
||||
Let $Y = X_1/X_2$. Then the pdf of $Y$ is, for $y > 0$, $$p_Y(y) = \int_0^{\infty} x p_{X_1} (xy) p_{X_2}(x) dx = \frac{4}{\pi^2} \int_0^\infty \frac{x}{(1+x^2 y^2)(1+x^2)}dx$$ $$=\frac{2}{\pi^2 (y^2-1)} \left[\log \left( \frac{1+x^2 y^2}{1+x^2}\right) \right]_{x=0}^{\infty} = \frac{2}{\pi^2} \frac{\log(y^2)}{y^2-1} = \frac{4}{\pi^2} \frac{\log(y)}{y^2-1}.$$
|
||||
Since $X_1$ and $X_2$ are equally likely to be the larger of the two, we have $P(Y < 1) = 1/2$. Thus $$\frac{1}{2} = \int_0^1 \frac{4}{\pi^2} \frac{\log(y)}{y^2-1} dy.$$ This is equivalent to $$\frac{\pi^2}{8} = \int_0^1 \frac{-\log(y)}{1-y^2} dy = -\int_0^1 \log(y) (1+y^2+y^4 + \cdots) dy = \sum_{k=0}^\infty \frac{1}{(2k+1)^2},$$ which, as others have pointed out, implies $\zeta(2) = \pi^2/6$.
|
||||
|
||||
Hans Lundmark
|
||||
This is not really an answer, but rather a long comment prompted by David Speyer's answer. The proof that David gives seems to be the one in How to compute $\sum 1/n^2$ by solving triangles by Mikael Passare, although that paper uses a slightly different way of seeing that the area of the region $U_0$ (in Passare's notation) bounded by the positive axes and the curve $e^{-x}+e^{-y}=1$, $$\int_0^{\infty} -\ln(1-e^{-x}) dx,$$ is equal to $\sum_{n\ge 1} \frac{1}{n^2}$.
|
||||
This brings me to what I really wanted to mention, namely another curious way to see why $U_0$ has that area; I learned this from Johan Wästlund. Consider the region $D_N$ illustrated below for $N=8$:
|
||||
A shape with area = sum of reciprocal squares
|
||||
Although it's not immediately obvious, the area of $D_N$ is $\sum_{n=1}^N \frac{1}{n^2}$. Proof: The area of $D_1$ is 1. To get from $D_N$ to $D_{N+1}$ one removes the boxes along the top diagonal, and adds a new leftmost column of rectangles of width $1/(N+1)$ and heights $1/1,1/2,\ldots,1/N$, plus a new bottom row which is the "transpose" of the new column, plus a square of side $1/(N+1)$ in the bottom left corner. The $k$th rectangle from the top in the new column and the $k$th rectangle from the left in the new row (not counting the square) have a combined area which exactly matches the $k$th box in the removed diagonal: $$ \frac{1}{k} \frac{1}{N+1} + \frac{1}{N+1} \frac{1}{N+1-k} = \frac{1}{k} \frac{1}{N+1-k}. $$ Thus the area added in the process is just that of the square, $1/(N+1)^2$. Q.E.D.
|
||||
(Apparently this shape somehow comes up in connection with the "random assignment problem", where there's an expected value of something which turns out to be $\sum_{n=1}^N \frac{1}{n^2}$.)
|
||||
Now place $D_N$ in the first quadrant, with the lower left corner at the origin. Letting $N\to\infty$ gives nothing but the region $U_0$: for large $N$ and for $0<\alpha<1$, the upper corner of column number $\lceil \alpha N \rceil$ in $D_N$ lies at $$ (x,y) = \left( \sum_{n=\lceil (1-\alpha) N \rceil}^N \frac{1}{n}, \sum_{n=\lceil \alpha N \rceil}^N \frac{1}{n} \right) \sim \left(\ln\frac{1}{1-\alpha}, \ln\frac{1}{\alpha}\right),$$ hence (in the limit) on the curve $e^{-x}+e^{-y}=1$.
|
||||
|
||||
xpaul
|
||||
Note that $$ \frac{\pi^2}{\sin^2\pi z}=\sum_{n=-\infty}^{\infty}\frac{1}{(z-n)^2} $$ from complex analysis and that both sides are analytic everywhere except $n=0,\pm 1,\pm 2,\cdots$. Then one can obtain $$ \frac{\pi^2}{\sin^2\pi z}-\frac{1}{z^2}=\sum_{n=1}^{\infty}\frac{1}{(z-n)^2}+\sum_{n=1}^{\infty}\frac{1}{(z+n)^2}. $$ Now the right hand side is analytic at $z=0$ and hence $$\lim_{z\to 0}\left(\frac{\pi^2}{\sin^2\pi z}-\frac{1}{z^2}\right)=2\sum_{n=1}^{\infty}\frac{1}{n^2}.$$ Note $$\lim_{z\to 0}\left(\frac{\pi^2}{\sin^2\pi z}-\frac{1}{z^2}\right)=\frac{\pi^2}{3}.$$ Thus $$\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}.$$
|
||||
|
||||
Jack D'Aurizio
|
||||
Just as a curiosity, a one-line-real-analytic-proof I found by combining different ideas from this thread and this question:
|
||||
$$\begin{eqnarray*}\zeta(2)&=&\frac{4}{3}\sum_{n=0}^{+\infty}\frac{1}{(2n+1)^2}=\frac{4}{3}\int_{0}^{1}\frac{\log y}{y^2-1}dy\\&=&\frac{2}{3}\int_{0}^{1}\frac{1}{y^2-1}\left[\log\left(\frac{1+x^2 y^2}{1+x^2}\right)\right]_{x=0}^{+\infty}dy\\&=&\frac{4}{3}\int_{0}^{1}\int_{0}^{+\infty}\frac{x}{(1+x^2)(1+x^2 y^2)}dx\,dy\\&=&\frac{4}{3}\int_{0}^{1}\int_{0}^{+\infty}\frac{dx\, dz}{(1+x^2)(1+z^2)}=\frac{4}{3}\cdot\frac{\pi}{4}\cdot\frac{\pi}{2}=\frac{\pi^2}{6}.\end{eqnarray*}$$
|
||||
Update. By collecting pieces, I have another nice proof. By Euler's acceleration method or just an iterated trick like my $(1)$ here we get: $$ \zeta(2) = \sum_{n\geq 1}\frac{1}{n^2} = \color{red}{\sum_{n\geq 1}\frac{3}{n^2\binom{2n}{n}}}\tag{A}$$ and the last series converges pretty fast. Then we may notice that the last series comes out from a squared arcsine. That just gives another proof of $ \zeta(2)=\frac{\pi^2}{6}$.
|
||||
A proof of the identity $$\sum_{n\geq 0}\frac{1}{(2n+1)^2}=\frac{\pi}{2}\sum_{k\geq 0}\frac{(-1)^k}{2k+1}=\frac{\pi}{2}\cdot\frac{\pi}{4}$$ is also hidden in tired's answer here. For short, the integral $$ I=\int_{-\infty}^{\infty}e^y\left(\frac{e^y-1}{y^2}-\frac{1}{y}\right)\frac{1}{e^{2y}+1}\,dy $$ is clearly real, so the imaginary part of the sum of residues of the integrand function has to be zero.
|
||||
Still another way (and a very efficient one) is to exploit the reflection formula for the trigamma function: $$\psi'(1-z)+\psi'(z)=\frac{\pi^2}{\sin^2(\pi z)}$$ immediately leads to: $$\frac{\pi^2}{2}=\psi'\left(\frac{1}{2}\right)=\sum_{n\geq 0}\frac{1}{\left(n+\frac{1}{2}\right)^2}=4\sum_{n\geq 0}\frac{1}{(2n+1)^2}=3\,\zeta(2).$$
|
||||
2018 update. We may consider that $\mathcal{J}=\int_{0}^{+\infty}\frac{\arctan x}{1+x^2}\,dx = \left[\frac{1}{2}\arctan^2 x\right]_0^{+\infty}=\frac{\pi^2}{8}$.
|
||||
On the other hand, by Feynman's trick or Fubini's theorem $$ \mathcal{J}=\int_{0}^{+\infty}\int_{0}^{1}\frac{x}{(1+x^2)(1+a^2 x^2)}\,da\,dx = \int_{0}^{1}\frac{-\log a}{1-a^2}\,da $$ and since $\int_{0}^{1}-\log(x)x^n\,dx = \frac{1}{(n+1)^2}$, by expanding $\frac{1}{1-a^2}$ as a geometric series we have $$ \frac{\pi^2}{8}=\mathcal{J}=\sum_{n\geq 0}\frac{1}{(2n+1)^2}. $$
|
||||
|
||||
Andrey Rekalo
|
||||
Here is a complex-analytic proof.
|
||||
For $z\in D=\mathbb{C}\backslash${$0,1$}, let
|
||||
$$R(z)=\sum\frac{1}{\log^2 z}$$
|
||||
where the sum is taken over all branches of the logarithm. Each point in $D$ has a neighbourhood on which the branches of $\log(z)$ are analytic. Since the series converges uniformly away from $z=1$, $R(z)$ is analytic on $D$.
|
||||
Now a few observations:
|
||||
(i) Each term of the series tends to $0$ as $z\to0$. Thanks to the uniform convergence this implies that the singularity at $z=0$ is removable and we can set $R(0)=0$.
|
||||
(ii) The only singularity of $R$ is a double pole at $z=1$ due to the contribution of the principal branch of $\log z$. Moreover, $\lim_{z\to1}(z-1)^2R(z)=1$.
|
||||
(iii) $R(1/z)=R(z)$.
|
||||
By (i) and (iii) $R$ is meromorphic on the extended complex plane, therefore it is rational. By (ii) the denominator of $R(z)$ is $(z-1)^2$. Since $R(0)=R(\infty)=0$, the numerator has the form $az$. Then (ii) implies $a=1$, so that $$R(z)=\frac{z}{(z-1)^2}.$$
|
||||
Now, setting $z=e^{2\pi i w}$ yields $$\sum\limits_{n=-\infty}^{\infty}\frac{1}{(w-n)^2}=\frac{\pi^2}{\sin^2(\pi w)}$$ which implies that $$\sum\limits_{k=0}^{\infty}\frac{1}{(2k+1)^2}=\frac{\pi^2}{8},$$ and the identity $\zeta(2)=\pi^2/6$ follows.
|
||||
The proof is due to T. Marshall (American Mathematical Monthly, Vol. 117(4), 2010, P. 352).
|
||||
|
||||
David Speyer
|
||||
In response to a request here: Compute $\oint z^{-2k} \cot (\pi z) dz$ where the integral is taken around a square of side $2N+1$. Routine estimates show that the integral goes to $0$ as $N \to \infty$.
|
||||
Now, let's compute the integral by residues. At $z=0$, the residue is $\pi^{2k-1} q$, where $q$ is some rational number coming from the power series for $\cot$. For example, if $k=1$, then we get $- \pi/3$.
|
||||
At $m \pi$, for $m \neq 0$, the residue is $z^{-2k} \pi^{-1}$. So $$\pi^{-1} \lim_{N \to \infty} \sum_{-N \leq m \leq N\ m \neq 0} m^{-2k} + \pi^{2k-1} q=0$$ or $$\sum_{m=1}^{\infty} m^{-2k} = -\pi^{2k} q/2$$ as desired. In particular, $\sum m^{-2} = - (\pi^2/3)/2 = \pi^2/6$.
|
||||
Common variants: We can replace $\cot$ with $\tan$, with $1/(e^{2 \pi i z}-1)$, or with similar formulas.
|
||||
This is reminiscent of Qiaochu's proof but, rather than actually establishing the relation $\pi^{-1} \cot(\pi z) = \sum (z-n)^{-1}$, one simply establishes that both sides contribute the same residues to a certain integral.
|
||||
|
||||
Derek Jennings
|
||||
Another variation. We make use of the following identity (proved at the bottom of this note):
|
||||
$$\sum_{k=1}^n \cot^2 \left( \frac {2k-1}{2n} \frac{\pi}{2} \right) = 2n^2 - n. \quad (1)$$
|
||||
Now $1/\theta > \cot \theta > 1/\theta - \theta/3 > 0$ for $0< \theta< \pi/2 < \sqrt{3}$ and so $$ 1/\theta^2 - 2/3 < \cot^2 \theta < 1/\theta^2. \quad (2)$$
|
||||
With $\theta_k = (2k-1)\pi/4n,$ summing the inequalities $(2)$ from $k=1$ to $n$ we obtain
|
||||
$$2n^2 - n < \sum_{k=1}^n \left( \frac{2n}{2k-1}\frac{2}{\pi} \right)^2 < 2n^2 - n + 2n/3.$$
|
||||
Hence
|
||||
$$\frac{\pi^2}{16}\frac{2n^2-n}{n^2} < \sum_{k=1}^n \frac{1}{(2k-1)^2} < \frac{\pi^2}{16}\frac{2n^2-n/3}{n^2}.$$
|
||||
Taking the limit as $n \rightarrow \infty$ we obtain
|
||||
$$ \sum_{k=1}^\infty \frac{1}{(2k-1)^2} = \frac{\pi^2}{8},$$
|
||||
from which the result for $\sum_{k=1}^\infty 1/k^2$ follows easily.
|
||||
To prove $(1)$ we note that
|
||||
$$ \cos 2n\theta = \text{Re}(\cos\theta + i \sin\theta)^{2n} = \sum_{k=0}^n (-1)^k {2n \choose 2k}\cos^{2n-2k}\theta\sin^{2k}\theta.$$
|
||||
Therefore
|
||||
$$\frac{\cos 2n\theta}{\sin^{2n}\theta} = \sum_{k=0}^n (-1)^k {2n \choose 2k}\cot^{2n-2k}\theta.$$
|
||||
And so setting $x = \cot^2\theta$ we note that
|
||||
$$f(x) = \sum_{k=0}^n (-1)^k {2n \choose 2k}x^{n-k}$$
|
||||
has roots $x_j = \cot^2 (2j-1)\pi/4n,$ for $j=1,2,\ldots,n,$ from which $(1)$ follows since ${2n \choose 2n-2} = 2n^2-n.$
|
||||
|
||||
xpaul
|
||||
A short way to get the sum is to use Fourier's expansion of $x^2$ in $x\in(-\pi,\pi)$. Recall that Fourier's expansion of $f(x)$ is $$ \tilde{f}(x)=\frac{1}{2}a_0+\sum_{n=1}^\infty(a_n\cos nx+b_n\sin nx), x\in(-\pi,\pi)$$ where $$ a_0=\frac{2}{\pi}\int_{-\pi}^{\pi}f(x)\;dx, a_n=\frac{2}{\pi}\int_{-\pi}^{\pi}f(x)\cos nx\; dx, b_n=\frac{2}{\pi}\int_{-\pi}^{\pi}f(x)\sin nx\; dx, n=1,2,3,\cdots $$ and $$ \tilde{f}(x)=\frac{f(x-0)+f(x+0)}{2}. $$ Easy calculation shows $$ x^2=\frac{\pi^2}{3}+4\sum_{n=1}^\infty(-1)^n\frac{\cos nx}{n^2}, x\in[-\pi,\pi]. $$ Letting $x=\pi$ in both sides gives $$ \sum_{n=1}^\infty\frac{1}{n^2}=\frac{\pi^2}{6}.$$
|
||||
Another way to get the sum is to use Parseval's Identity for Fourier's expansion of $x$ in $(-\pi,\pi)$. Recall that Parseval's Identity is $$ \int_{-\pi}^{\pi}|f(x)|^2dx=\frac{1}{2}a_0^2+\sum_{n=1}^\infty(a_n^2+b_n^2). $$ Note $$ x=2\sum_{n=1}^\infty(-1)^{n+1}\frac{\sin nx}{n}, x\in(-\pi,\pi). $$ Using Parseval's Identity gives $$ 4\sum_{n=1}^\infty\frac{1}{n^2}=\int_{-\pi}^{\pi}|x|^2dx$$ or $$ \sum_{n=1}^\infty\frac{1}{n^2}=\frac{\pi^2}{6}.$$
|
||||
|
||||
Tomás
|
||||
I like this one:
|
||||
Let $f\in Lip(S^{1})$, where $Lip(S^{1})$ is the space of Lipschitz functions on $S^{1}$. So its well defined the number for $k\in \mathbb{Z}$ (called Fourier series of $f$) $$\hat{f}(k)=\frac{1}{2\pi}\int \hat{f}(\theta)e^{-ik\theta}d\theta.$$
|
||||
By the inversion formula, we have $$f(\theta)=\sum_{k\in\mathbb{Z}}\hat{f}(k)e^{ik\theta}.$$
|
||||
Now take $f(\theta)=|\theta|$, $\theta\in [-\pi,\pi]$. Note that $f\in Lip(S^{1})$
|
||||
We have $$ \hat{f}(k) = \left\{ \begin{array}{rl} \frac{\pi}{2} &\mbox{ if $k=0$} \\ 0 &\mbox{ if $|k|\neq 0$ and $|k|$ is even} \\ -\frac{2}{k^{2}\pi} &\mbox{if $|k|\neq 0$ and $|k|$ is odd} \end{array} \right. $$
|
||||
Using the inversion formula, we have on $\theta=0$ that $$0=\sum_{k\in\mathbb{Z}}\hat{f}(k).$$
|
||||
Then,
|
||||
\begin{eqnarray} 0 &=& \frac{\pi}{2}-\sum_{k\in\mathbb{Z}\ |k|\ odd}\frac{2}{k^{2}\pi} \nonumber \\ &=& \frac{\pi}{2}-\sum_{k\in\mathbb{N}\ |k|\ odd}\frac{4}{k^{2}\pi} \nonumber \\ \end{eqnarray}
|
||||
This implies $$\sum_{k\in\mathbb{N}\ |k|\ odd}\frac{1}{k^{2}} =\frac{\pi^{2}}{8}$$
|
||||
If we multiply the last equation by $\frac{1}{2^{2n}}$ with $n=0,1,2,...$ ,we get $$\sum_{k\in\mathbb{N}\ |k|\ odd}\frac{1}{(2^{n}k)^{2}} =\frac{\pi^{2}}{2^{2n}8}$$
|
||||
Now $$\sum_{n=0,1,...}(\sum_{k\in\mathbb{N}\ |k|\ odd}\frac{1}{(2^{n}k)^{2}}) =\sum_{n=0,1,...}\frac{\pi^{2}}{2^{2n}8}$$
|
||||
The sum in the left is equal to: $\sum_{k\in\mathbb{N}}\frac{1}{k^{2}}$
|
||||
The sum in the right is equal to :$\frac{\pi^{2}}{6}$
|
||||
So we conclude: $$\sum_{k\in\mathbb{N}}\frac{1}{k^{2}}=\frac{\pi^{2}}{6}$$
|
||||
Note: This is problem 9, Page 208 from the boof of Michael Eugene Taylor - Partial Differential Equation Volume 1.
|
||||
|
||||
user91500
|
||||
Theorem: Let $\lbrace a_n\rbrace$ be a nonincreasing sequence of positive numbers such that $\sum a_n^2$ converges. Then both series $$s:=\sum_{n=0}^\infty(-1)^na_n,\,\delta_k:=\sum_{n=0}^\infty a_na_{n+k},\,k\in\mathbb N $$ converge. Morevere $\Delta:=\sum_{k=1}^\infty(-1)^{k-1}\delta_k$ also converges, and we have the formula $$\sum_{n=0}^\infty a_n^2=s^2+2\Delta.$$ Proof: Knopp. Konrad, Theory and Application of Infinite Series, page 323.
|
||||
If we let $a_n=\frac1{2n+1}$ in this theorem, then we have $$s=\sum_{n=0}^\infty(-1)^n\frac1{2n+1}=\frac\pi 4$$ $$\delta_k=\sum_{n=0}^\infty\frac1{(2n+1)(2n+2k+1)}=\frac1{2k}\sum_{n=0}^\infty\left(\frac1{2n+1}-\frac1{2n+2k+1}\right)=\frac{1}{2k}\left(1+\frac1 3+...+\frac1 {2k-1}\right)$$ Hence, $$\sum_{n=0}^\infty\frac1{(2n+1)^2}=\left(\frac\pi 4\right)^2+\sum_{k=1}^\infty\frac{(-1)^{k-1}}{k}\left(1+\frac1 3+...+\frac1 {2k-1}\right)=\frac{\pi^2}{16}+\frac{\pi^2}{16}=\frac{\pi^2}{8}$$ and now $$\zeta(2)=\frac4 3\sum_{n=0}^\infty\frac1{(2n+1)^2}=\frac{\pi^2}6.$$
|
||||
|
||||
Markus Scheuer
|
||||
Here's a proof based upon periods and the fact that $\zeta(2)$ and $\frac{\pi^2}{6}$ are periods forming an accessible identity.
|
||||
The definition of periods below and the proof is from the fascinating introductory survey paper about periods by M. Kontsevich and D. Zagier.
|
||||
Periods are defined as complex numbers whose real and imaginary parts are values of absolutely convergent integrals of rational functions with rational coefficient over domains in $\mathbb{R}^n$ given by polynomial inequalities with rational coefficients.
|
||||
The set of periods is therefore a countable subset of the complex numbers. It contains the algebraic numbers, but also many of famous transcendental constants.
|
||||
In order to show the equality $\zeta(2)=\frac{\pi^2}{6}$ we have to show that both are periods and that $\zeta(2)$ and $\frac{\pi^2}{6}$ form a so-called accessible identity.
|
||||
First step of the proof: $\zeta(2)$ and $\pi$ are periods
|
||||
There are a lot of different proper representations of $\pi$ showing that this constant is a period. In the referred paper above following expressions (besides others) of $\pi$ are stated:
|
||||
\begin{align*} \pi= \iint \limits_{x^2+y^2\leq 1}dxdy=\int_{-\infty}^{\infty}\frac{dx}{1+x^2} \end{align*}
|
||||
showing that $\pi$ is a period. The known representation
|
||||
\begin{align*} \zeta(2)=\iint_{0<x<y<1} \frac{dxdy}{(1-x)y} \end{align*}
|
||||
shows that $\zeta(2)$ is also a period.
|
||||
$$ $$
|
||||
Second step: $\zeta(2)$ and $\frac{\pi^2}{6}$ form an accessible identity.
|
||||
An accessible identity between two periods $A$ and $B$ is given, if we can transform the integral representation of period $A$ by application of the three rules: Additivity (integrand and domain), Change of variables and Newton-Leibniz formula to the integral representation of period $B$.
|
||||
This implies equality of the periods and the job is done.
|
||||
In order to show that $\zeta(2)$ and $\frac{\pi^2}{6}$ are accessible identities we start with the integral $I$
|
||||
$$I=\int_{0}^{1}\int_{0}^{1}\frac{1}{1-xy}\frac{dxdy}{\sqrt{xy}}$$
|
||||
Expanding $1/(1-xy)$ as a geometric series and integrating term-by-term,
|
||||
we find that
|
||||
$$I=\sum_{n=0}^{\infty}\left(n+\frac{1}{2}\right)^{-2}=(4-1)\zeta(2),$$
|
||||
providing another period representation of $\zeta(2)$.
|
||||
Changing variables:
|
||||
$$x=\xi^2\frac{1+\eta^2}{1+\xi^2},\qquad\qquad y=\eta^2\frac{1+\xi^2}{1+\eta^2}$$
|
||||
with Jacobian $\left|\frac{\partial(x,y)}{\partial(\xi,\eta)}\right|=\frac{4\xi\eta(1-\xi^2\eta^2)}{(1+\xi^2)(1+\eta^2)} =4\frac{(1-xy)\sqrt{xy}}{(1+\xi^2)(1+\eta^2)}$, we find
|
||||
$$I=4\iint_{0<\eta,\xi\leq 1}\frac{d\xi}{1+\xi^2}\frac{d\eta}{1+\eta^2} =2\int_{0}^{\infty}\frac{d\xi}{1+\xi^2}\int_{0}^{\infty}\frac{d\eta}{1+\eta^2},$$
|
||||
the last equality being obtained by considering the involution $(\xi,\eta) \mapsto (\xi^{-1},\eta^{-1})$ and comparing this with the last integral representation of $\pi$ above we obtain: $$I=\frac{\pi^2}{2}$$
|
||||
So, we have shown that $\frac{\pi^2}{6}$ and $\zeta(2)$ are accessible identities and equality follows.
|
||||
|
||||
🐢
|
||||
|
||||
I Want To Remain Anonymous
|
||||
As taken from my upcoming textbook:
|
||||
There is yet another solution to the Basel problem as proposed by Ritelli (2013). His approach is similar to the one by Apostol (1983), where he arrives at
|
||||
$$\sum_{n\geq1}\frac{1}{n^2}=\frac{\pi^2}{6}\tag1$$
|
||||
by evaluating the double integral
|
||||
$$\int_0^1\int_0^1\dfrac{\mathrm{d}x\,\mathrm{d}y}{1-xy}.\tag2$$
|
||||
Ritelli evaluates in this case the definite integral shown in $(4)$. The starting point comes from realizing that $(1)$ is equivalent to
|
||||
$$\sum_{n\geq0}\frac{1}{(2n+1)^2}=\frac{\pi^2}{8}\tag3$$
|
||||
To evaluate the above sum we consider the definite integral
|
||||
$$\int_0^\infty\int_0^\infty\frac{\mathrm{d}x\,\mathrm{d}y}{(1+y)(1+x^2y)}.\tag4$$
|
||||
We evaluate $(4)$ first with respect to $x$ and then to $y$
|
||||
$$\begin{align} \int_0^\infty\left(\frac{1}{1+y}\int_0^\infty\frac{\mathrm{d}x}{1+x^2y}\right)\mathrm{d}y &=\int_0^\infty\left(\frac{1}{1+y}\left[\frac{\tan^{-1}(\sqrt{y}\,x)}{\sqrt{y}}\right]_{x=0}^{x=\infty}\right)\mathrm{d}y\\ &=\frac\pi2\int_0^\infty\frac{\mathrm{d}y}{\sqrt{y}(1+y)}\\ &=\frac\pi2\int_0^\infty\frac{2u}{u(1+u^2)}\mathrm{d}u=\frac{\pi^2}{2},\tag5 \end{align}$$
|
||||
where we used the substitution $y\leadsto u^2$ in the last step. If we reverse the order of integration one gets
|
||||
$$\begin{align} \int_0^\infty\left(\int_0^\infty\frac{\mathrm{d}y}{(1+y)(1+x^2y)}\right)\mathrm{d}x&=\int_0^\infty\frac{1}{1-x^2}\left(\int_0^\infty\left(\frac{1}{1+y}-\frac{x^2}{1+x^2y}\right)\mathrm{d}y\right)\mathrm{d}x\\ &=\int_0^\infty\frac{1}{1-x^2}\ln\frac1{x^2}\mathrm{d}x=2\int_0^\infty\frac{\ln x}{x^2-1}\mathrm{d}x.\tag6 \end{align}$$
|
||||
Hence since $(5)$ and $(6)$ are the same, we have
|
||||
$$\int_0^\infty\frac{\ln x}{x^2-1}\mathrm{d}x=\frac{\pi^2}{4}.\tag7$$
|
||||
Furthermore
|
||||
$$\begin{align} \int_0^\infty\frac{\ln x}{x^2-1}\mathrm{d}x&=\int_0^1\frac{\ln x}{x^2-1}\mathrm{d}x+\int_1^\infty\frac{\ln x}{x^2-1}\mathrm{d}x\\ &=\int_0^1\frac{\ln x}{x^2-1}\mathrm{d}x+\int_0^1\frac{\ln u}{u^2-1}\mathrm{d}u,\tag8 \end{align}$$
|
||||
where we used the substitution $x\leadsto1/u$. Combining $(7)$ and $(8)$ yields
|
||||
$$\int_0^1\frac{\ln x}{x^2-1}\mathrm{d}x=\frac{\pi^2}{8}.\tag{9}$$
|
||||
By expanding the denominator of the integrand in $(10)$ into a geometric series and using the Monotone Convergence Theorem,
|
||||
$$\int_0^1\frac{\ln x}{x^2-1}\mathrm{d}x=\int_0^1\frac{-\ln x}{1-x^2}\mathrm{d}x=\sum_{n\ge0}\int_0^1(-x^{2n}\ln x)\mathrm{d}x.\tag{10}$$
|
||||
Using integration by parts one can see that
|
||||
$$\int_0^1(-x^{2n}\ln x)\mathrm{d}x=\left[-\frac{x^{2n+1}}{2n+1}\ln x\right]^1_0+\int_0^1\frac{x^{2n}}{2n+1}\mathrm{d}x=\frac{1}{(2n+1)^2}\tag{11}$$
|
||||
Hence from $(10)$, and $(11)$
|
||||
$$\int_0^1\frac{\ln x}{x^2-1}\mathrm{d}x=\sum_{n\geq0}\frac{1}{(2n+1)^2},\tag{12}$$
|
||||
which finishes the proof. $$\tag*{$\square$}$$
|
||||
References:
|
||||
Daniele Ritelli (2013), Another Proof of $\zeta(2)=\frac{\pi^2}{6}$ Using Double Integrals, The American Mathematical Monthly, Vol. 120, No. 7, pp. 642-645
|
||||
T. Apostol (1983), A proof that Euler missed: Evaluating $\zeta(2)$ the easy way, Math. Intelligencer 5, pp. 59-60, available at http://dx.doi.org/10.1007/BF03026576.
|
||||
|
||||
Eugene Shvarts
|
||||
This popped up in some reading I'm doing for my research, so I thought I'd contribute! It's a more general twist on the usual pointwise-convergent Fourier series argument.
|
||||
Consider the eigenvalue problem for the negative Laplacian $\mathcal L$ on $[0,1]$ with Dirichlet boundary conditions; that is, $\mathcal L f:=-f_n'' = \lambda_n f_n$ with $f_n(0) = f_n(1) = 0$. Through inspection we can find that the admissible eigenvalues are $\lambda_n = n^2\pi^2$ for $n=1,2,\ldots$
|
||||
One can verify that the integral operator $\mathcal Gf(x) = \int_0^1 G(x,y)f(y)\,dy$, where $$G(x,y) = \min(x,y) - xy = \frac{1}{2}\left( -|x-y| + x(1-y) + y(1-x) \right)~~,$$ inverts the negative Laplacian, in the sense that $\mathcal L \mathcal G f = \mathcal G \mathcal L f = f$ on the admissible class of functions (twice weakly differentiable, satisfying the boundary conditions). That is, $G$ is the Green's function for the Dirichlet Laplacian. Because $\mathcal G$ is a self-adjoint, compact operator, we can form an orthonormal basis for $L^2([0,1])$ from its eigenfunctions, and so may express its trace in two ways: $$ \sum_n <f_n,\mathcal G f_n> = \sum_n \frac{1}{\lambda_n} $$and $$\sum_n <f_n,\mathcal G f_n> = \int_0^1 \sum_n f_n(x) <G(x,\cdot),f_n>\,dx = \int_0^1 G(x,x)\,dx~~.$$
|
||||
The latter quantity is $$ \int_0^1 x(1-x)\,dx = \frac 1 2 - \frac 1 3 = \frac 1 6~~.$$
|
||||
Hence, we have that $$\sum_n \frac 1 {n^2\pi^2} = \frac 1 6~~\text{, or}~~ \sum_n \frac 1 {n^2} = \frac {\pi^2} 6~~.$$
|
||||
|
||||
Markus Scheuer
|
||||
Here is Euler's Other Proof by Gerald Kimble
|
||||
> $$
|
||||
\begin{align*} \frac{\pi^2}{6}&=\frac{4}{3}\frac{(\arcsin 1)^2}{2}\\ &=\frac{4}{3}\int_0^1\frac{\arcsin x}{\sqrt{1-x^2}}\,dx\\ &=\frac{4}{3}\int_0^1\frac{x+\sum_{n=1}^{\infty}\frac{(2n-1)!!}{(2n)!!}\frac{x^{2n+1}}{2n+1}}{\sqrt{1-x^2}}\,dx\\ &=\frac{4}{3}\int_0^1\frac{x}{\sqrt{1-x^2}}\,dx +\frac{4}{3}\sum_{n=1}^{\infty}\frac{(2n-1)!!}{(2n)!!(2n+1)}\int_0^1x^{2n}\frac{x}{\sqrt{1-x^2}}\,dx\\ &=\frac{4}{3}+\frac{4}{3}\sum_{n=1}^{\infty}\frac{(2n-1)!!}{(2n)!!(2n+1)}\left[\frac{(2n)!!}{(2n+1)!!}\right]\\ &=\frac{4}{3}\sum_{n=0}^{\infty}\frac{1}{(2n+1)^2}\\ &=\frac{4}{3}\left(\sum_{n=1}^{\infty}\frac{1}{n^2}-\frac{1}{4}\sum_{n=1}^{\infty}\frac{1}{n^2}\right)\\ &=\sum_{n=1}^{\infty}\frac{1}{n^2} \end{align*}
|
||||
$$
|
||||
|
||||
B_Scheiner
|
||||
Consider the function $\pi \cot(\pi z)$ which has poles at $z=\pm n$ where n is an integer. Using the L'hopital rule you can see that the residue at these poles is 1.
|
||||
Now consider the integral $\int_{\gamma_N} \frac{\pi\cot(\pi z)}{z^2} dz$ where the contour $\gamma_N$ is the rectangle with corners given by ±(N + 1/2) ± i(N + 1/2) so that the contour avoids the poles of $\cot(\pi z)$. The integral is bouond in the following way: $\int_{\gamma_N} |\frac{\pi\cot(\pi z)}{z^2} |dz\le Max |(\frac{\pi\cot(\pi z)}{z^2}) | Length(\gamma_N)$. It can easily be shown that on the contour $\gamma_N$ that $\pi \cot(\pi z)< M$ where M is some constant. Then we have
|
||||
$\int_{\gamma_N} |\frac{\pi\cot(\pi z)}{z^2} |dz\le M Max |\frac{1}{z^2} | Length(\gamma_N) = (8N+4) \frac{M}{\sqrt{2(1/2+N)^2}^2}$
|
||||
where (8N+4) is the lenght of the contour and $\sqrt{2(1/2+N)^2}$ is half the diagonal of $\gamma_N$. In the limit that N goes to infinity the integral is bound by 0 so we have $\int_{\gamma_N} \frac{\pi\cot(\pi z)}{z^2} dz =0$
|
||||
by the cauchy residue theorem we have 2πiRes(z = 0) + 2πi$\sum$Residues(z$\ne$ 0) = 0. At z=0 we have Res(z=0)=$-\frac{\pi^2}{3}$, and $Res (z=n)=\frac{1}{n^2}$ so we have
|
||||
$2\pi iRes(z = 0) + 2\pi i\sum Residues(z\ne 0) = -\frac{\pi^2}{3}+2\sum_{1}^{\infty} \frac{1}{n^2} =0$
|
||||
Where the 2 in front of the residue at n is because they occur twice at +/- n.
|
||||
We now have the desired result $\sum_{1}^{\infty} \frac{1}{n^2}=\frac{\pi^2}{6}$.
|
||||
|
||||
Meadara
|
||||
I saw this proof in an extract of the College Mathematics Journal.
|
||||
Consider the Integeral : $I = \int_0^{\pi/2}\ln(2\cos x)dx$
|
||||
From $2\cos(x) = e^{ix} + e^{-ix}$ , we have:
|
||||
$$\int_0^{\pi/2}\ln\left(e^{ix} + e^{-ix}\right)dx = \int_0^{\pi/2}\ln\left(e^{ix}(1 + e^{-2ix})\right)dx=\int_0^{\pi/2}ixdx + \int_0^{\pi/2}\ln(1 + e^{-2ix})dx$$
|
||||
The Taylor series expansion of $\ln(1+x)=x -\frac{x^2}{2} +\frac{x^3}{3}-\cdots$
|
||||
Thus , $\ln(1+e^{-2ix}) = e^{-2ix}- \frac{e^{-4ix}}{2} + \frac{e^{-6ix}}{3} - \cdots $, then for $I$ :
|
||||
$$I = \frac{i\pi^2}{8}+\left[-\frac{e^{-2ix}}{2i}+\frac{e^{-4ix}}{2\cdot 4i}-\frac{e^{-6ix}}{3\cdot 6i}-\cdots\right]_0^\frac{\pi}{2}$$
|
||||
$$I = \frac{i\pi^2}{8}-\frac{1}{2i}\left[\frac{e^{-2ix}}{1^2}-\frac{e^{-4ix}}{2^2}+\frac{e^{-6ix}}{3^2}-\cdots\right]_0^\frac{\pi}{2}$$
|
||||
By evaluating we get something like this..
|
||||
$$I = \frac{i\pi^2}{8}-\frac{1}{2i}\left[\frac{-2}{1^2}-\frac{0}{2^2}+\frac{-2}{3^2}-\cdots\right]_0^\frac{\pi}{2}$$
|
||||
Hence
|
||||
$$\int_0^{\pi/2}\ln(2\cos x)dx=\frac{i\pi^2}{8}-i\sum_{k=0}^\infty \frac{1}{(2k+1)^2}$$
|
||||
So now we have a real integral equal to an imaginary number, thus the value of the integral should be zero.
|
||||
Thus, $\sum_{k=0}^\infty \frac{1}{(2k+1)^2}=\frac{\pi^2}{8}$
|
||||
But let $\sum_{k=0}^\infty \frac{1}{k^2}=E$ .We get $\sum_{k=0}^\infty \frac{1}{(2k+1)^2}=\frac{3}{4} E$
|
||||
And as a result $$\sum_{k=0}^\infty \frac{1}{k^2} = \frac{\pi^2}{6}$$
|
||||
|
||||
dustin
|
||||
I have another method as well. From skimming the previous solutions, I don't think it is a duplicate of any of them
|
||||
In Complex analysis, we learn that $\sin(\pi z) = \pi z\Pi_{n=1}^{\infty}\Big(1 - \frac{z^2}{n^2}\Big)$ which is an entire function with simple zer0s at the integers. We can differentiate term wise by uniform convergence. So by logarithmic differentiation we obtain a series for $\pi\cot(\pi z)$. $$ \frac{d}{dz}\ln(\sin(\pi z)) = \pi\cot(\pi z) = \frac{1}{z} - 2z\sum_{n=1}^{\infty}\frac{1}{n^2 - z^2} $$ Therefore, $$ -\sum_{n=1}^{\infty}\frac{1}{n^2 - z^2} = \frac{\pi\cot(\pi z) - \frac{1}{z}}{2z} $$ We can expand $\pi\cot(\pi z)$ as $$ \pi\cot(\pi z) = \frac{1}{z} - \frac{\pi^2}{3}z - \frac{\pi^4}{45}z^3 - \cdots $$ Thus, \begin{align} \frac{\pi\cot(\pi z) - \frac{1}{z}}{2z} &= \frac{- \frac{\pi^2}{3}z - \frac{\pi^4}{45}z^3-\cdots}{2z}\\ -\sum_{n=1}^{\infty}\frac{1}{n^2 - z^2}&= -\frac{\pi^2}{6} - \frac{\pi^4}{90}z^2 - \cdots\\ -\lim_{z\to 0}\sum_{n=1}^{\infty}\frac{1}{n^2 - z^2}&= \lim_{z\to 0}\Big(-\frac{\pi^2}{6} - \frac{\pi^4}{90}z^2 - \cdots\Big)\\ -\sum_{n=1}^{\infty}\frac{1}{n^2}&= -\frac{\pi^2}{6}\\ \sum_{n=1}^{\infty}\frac{1}{n^2}&= \frac{\pi^2}{6} \end{align}
|
||||
|
||||
Elias
|
||||
See evaluations of Riemann Zeta Function $\zeta(2)=\sum_{n=1}^\infty\frac{1}{n^2}$ in mathworld.wolfram.com and a solution by in D. P. Giesy in Mathematics Magazine:
|
||||
D. P. Giesy, Still another elementary proof that $\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}$, Math. Mag. 45 (1972) 148-149.
|
||||
Unfortunately I did not get a link to this article. But there is a link to a note from Robin Chapman seems to me a variation of proof's Giesy.
|
||||
|
||||
barto
|
||||
Applying the usual trick 1 transforming a series to an integral, we obtain
|
||||
$$\sum_{n=1}^\infty\frac1{n^2}=\int_0^1\int_0^1\frac{dxdy}{1-xy}$$
|
||||
where we use the Monotone Convergence Theorem to integrate term-wise.
|
||||
Then there's this ingenious change of variables 2, which I learned from Don Zagier during a lecture, and which he in turn got from a colleague:
|
||||
$$(x,y)=\left(\frac{\cos v}{\cos u},\frac{\sin u}{\sin v}\right),\quad0\leq u\leq v\leq \frac\pi2$$
|
||||
One verifies that it is bijective between the rectangle $[0,1]^2$ and the triangle $0\leq u\leq v\leq \frac\pi2$, and that its Jacobian determinant is precisely $1-x^2y^2$, which means $\frac1{1-x^2y^2}$ would be a neater integrand. For the moment, we have found
|
||||
$$J=\int_0^1\int_0^1\frac{dxdy}{1-x^2y^2}=\frac{\pi^2}8$$ (the area of the triangular domain in the $(u,v)$ plane).
|
||||
There are two ways to transform $\int\frac1{1-xy}$ into something $\int\frac1{1-x^2y^2}$ish:
|
||||
Manipulate $S=\sum_{n=1}^\infty\frac1{n^2}$: We have $\sum_{n=1}^\infty\frac1{(2n)^2}=\frac14S$ so $\sum_{n=0}^\infty\frac1{(2n+1)^2}=\frac34S$. Applying the series-integral transformation, we get $\frac34S=J$ so $$S=\frac{\pi^2}6$$
|
||||
Manipulate $I=\int_0^1\int_0^1\frac{dxdy}{1-xy}$: Substituting $(x,y)\leftarrow(x^2,y^2)$ we have $I=\int_0^1\int_0^1\frac{4xydxdy}{1-x^2y^2}$ so $$J=\int_0^1\int_0^1\frac{dxdy}{1-x^2y^2}=\int_0^1\int_0^1\frac{(1+xy-xy)dxdy}{1-x^2y^2}=I-\frac14I$$ whence $$I=\frac43J=\frac{\pi^2}6$$
|
||||
(It may be seen that they are essentially the same methods.)
|
||||
After looking at the comments it seems that this looks a lot like Proof 2 in the article by R. Chapman.
|
||||
See also: Multiple Integral $\int\limits_0^1\!\!\int\limits_0^1\!\!\int\limits_0^1\!\!\int\limits_0^1\frac1{1-xyuv}\,dx\,dy\,du\,dv$
|
||||
1 See e.g. Proof 1 in Chapman's article.
|
||||
2 It may have been a different one; maybe as in the above article. Either way, the idea to do something trigonometric was not mine.
|
||||
|
||||
Asier Calbet
|
||||
The sum can be written as the integral: $$\int_0^{\infty} \frac{x}{e^x-1} dx $$ This integral can be evaluated using a rectangular contour from 0 to $\infty$ to $\infty + \pi i$ to $ 0$ .
|
||||
|
||||
John M. Campbell
|
||||
There is a simple way of proving that $\sum_{n=1}^{\infty}\frac{1}{n^2} = \frac{\pi^2}{6}$ using the following well-known series identity: $$\left(\sin^{-1}(x)\right)^{2} = \frac{1}{2}\sum_{n=1}^{\infty}\frac{(2x)^{2n}}{n^2 \binom{2n}{n}}.$$ From the above equality, we have that $$x^2 = \frac{1}{2}\sum_{n=1}^{\infty}\frac{(2 \sin(x))^{2n}}{n^2 \binom{2n}{n}},$$ and we thus have that: $$\int_{0}^{\pi} x^2 dx = \frac{\pi^3}{12} = \frac{1}{2}\sum_{n=1}^{\infty}\frac{\int_{0}^{\pi} (2 \sin(x))^{2n} dx}{n^2 \binom{2n}{n}}.$$ Since $$\int_{0}^{\pi} \left(\sin(x)\right)^{2n} dx = \frac{\sqrt{\pi} \ \Gamma\left(n + \frac{1}{2}\right)}{\Gamma(n+1)},$$ we thus have that: $$\frac{\pi^3}{12} = \frac{1}{2}\sum_{n=1}^{\infty}\frac{ 4^{n} \frac{\sqrt{\pi} \ \Gamma\left(n + \frac{1}{2}\right)}{\Gamma(n+1)} }{n^2 \binom{2n}{n}}.$$ Simplifying the summand, we have that $$\frac{\pi^3}{12} = \frac{1}{2}\sum_{n=1}^{\infty}\frac{\pi}{n^2},$$ and we thus have that $\sum_{n=1}^{\infty}\frac{1}{n^2} = \frac{\pi^2}{6}$ as desired.
|
||||
`
|
||||
|
|
@ -28,6 +28,7 @@ import (
|
|||
"github.com/gohugoio/hugo/cache/dynacache"
|
||||
"github.com/gohugoio/hugo/common/hashing"
|
||||
"github.com/gohugoio/hugo/common/hugio"
|
||||
"github.com/gohugoio/hugo/common/types"
|
||||
"github.com/gohugoio/hugo/internal/warpc"
|
||||
"github.com/gohugoio/hugo/markup/converter/hooks"
|
||||
"github.com/gohugoio/hugo/markup/highlight"
|
||||
|
|
@ -199,13 +200,15 @@ func (ns *Namespace) Plainify(s any) (template.HTML, error) {
|
|||
|
||||
// ToMath converts a LaTeX string to math in the given format, default MathML.
|
||||
// This uses KaTeX to render the math, see https://katex.org/.
|
||||
func (ns *Namespace) ToMath(ctx context.Context, args ...any) (template.HTML, error) {
|
||||
func (ns *Namespace) ToMath(ctx context.Context, args ...any) (types.Result[template.HTML], error) {
|
||||
var res types.Result[template.HTML]
|
||||
|
||||
if len(args) < 1 {
|
||||
return "", errors.New("must provide at least one argument")
|
||||
return res, errors.New("must provide at least one argument")
|
||||
}
|
||||
expression, err := cast.ToStringE(args[0])
|
||||
if err != nil {
|
||||
return "", err
|
||||
return res, err
|
||||
}
|
||||
|
||||
katexInput := warpc.KatexInput{
|
||||
|
|
@ -214,23 +217,21 @@ func (ns *Namespace) ToMath(ctx context.Context, args ...any) (template.HTML, er
|
|||
Output: "mathml",
|
||||
MinRuleThickness: 0.04,
|
||||
ErrorColor: "#cc0000",
|
||||
ThrowOnError: true,
|
||||
},
|
||||
}
|
||||
|
||||
if len(args) > 1 {
|
||||
if err := mapstructure.WeakDecode(args[1], &katexInput.Options); err != nil {
|
||||
return "", err
|
||||
return res, err
|
||||
}
|
||||
}
|
||||
|
||||
// Make sure this isn't set by the client (for now).
|
||||
katexInput.Options.ThrowOnError = false
|
||||
|
||||
s := hashing.HashString(args...)
|
||||
key := "tomath/" + s[:2] + "/" + s[2:]
|
||||
fileCache := ns.deps.ResourceSpec.FileCaches.MiscCache()
|
||||
|
||||
return ns.cacheMath.GetOrCreate(key, func(string) (template.HTML, error) {
|
||||
v, err := ns.cacheMath.GetOrCreate(key, func(string) (template.HTML, error) {
|
||||
_, r, err := fileCache.GetOrCreate(key, func() (io.ReadCloser, error) {
|
||||
message := warpc.Message[warpc.KatexInput]{
|
||||
Header: warpc.Header{
|
||||
|
|
@ -248,6 +249,9 @@ func (ns *Namespace) ToMath(ctx context.Context, args ...any) (template.HTML, er
|
|||
if err != nil {
|
||||
return nil, err
|
||||
}
|
||||
if result.Header.Err != "" {
|
||||
return nil, errors.New(result.Header.Err)
|
||||
}
|
||||
return hugio.NewReadSeekerNoOpCloserFromString(result.Data.Output), nil
|
||||
})
|
||||
if err != nil {
|
||||
|
|
@ -258,6 +262,13 @@ func (ns *Namespace) ToMath(ctx context.Context, args ...any) (template.HTML, er
|
|||
|
||||
return template.HTML(s), err
|
||||
})
|
||||
|
||||
res = types.Result[template.HTML]{
|
||||
Value: v,
|
||||
Err: err,
|
||||
}
|
||||
|
||||
return res, nil
|
||||
}
|
||||
|
||||
// For internal use.
|
||||
|
|
|
|||
|
|
@ -14,6 +14,8 @@
|
|||
package transform_test
|
||||
|
||||
import (
|
||||
"fmt"
|
||||
"strings"
|
||||
"testing"
|
||||
|
||||
qt "github.com/frankban/quicktest"
|
||||
|
|
@ -150,6 +152,109 @@ disableKinds = ['page','rss','section','sitemap','taxonomy','term']
|
|||
`)
|
||||
}
|
||||
|
||||
func TestToMathError(t *testing.T) {
|
||||
t.Run("Default", func(t *testing.T) {
|
||||
t.Parallel()
|
||||
|
||||
files := `
|
||||
-- hugo.toml --
|
||||
disableKinds = ['page','rss','section','sitemap','taxonomy','term']
|
||||
-- layouts/index.html --
|
||||
{{ transform.ToMath "c = \\foo{a^2 + b^2}" }}
|
||||
`
|
||||
b, err := hugolib.TestE(t, files, hugolib.TestOptWarn())
|
||||
|
||||
b.Assert(err, qt.IsNotNil)
|
||||
b.Assert(err.Error(), qt.Contains, "KaTeX parse error: Undefined control sequence: \\foo")
|
||||
})
|
||||
|
||||
t.Run("Disable ThrowOnError", func(t *testing.T) {
|
||||
t.Parallel()
|
||||
|
||||
files := `
|
||||
-- hugo.toml --
|
||||
disableKinds = ['page','rss','section','sitemap','taxonomy','term']
|
||||
-- layouts/index.html --
|
||||
{{ $opts := dict "throwOnError" false }}
|
||||
{{ transform.ToMath "c = \\foo{a^2 + b^2}" $opts }}
|
||||
`
|
||||
b, err := hugolib.TestE(t, files, hugolib.TestOptWarn())
|
||||
|
||||
b.Assert(err, qt.IsNil)
|
||||
b.AssertFileContent("public/index.html", `#cc0000`) // Error color
|
||||
})
|
||||
|
||||
t.Run("Handle in template", func(t *testing.T) {
|
||||
t.Parallel()
|
||||
|
||||
files := `
|
||||
-- hugo.toml --
|
||||
disableKinds = ['page','rss','section','sitemap','taxonomy','term']
|
||||
-- layouts/index.html --
|
||||
{{ with transform.ToMath "c = \\foo{a^2 + b^2}" }}
|
||||
{{ with .Err }}
|
||||
{{ warnf "error: %s" . }}
|
||||
{{ else }}
|
||||
{{ . }}
|
||||
{{ end }}
|
||||
{{ end }}
|
||||
`
|
||||
b, err := hugolib.TestE(t, files, hugolib.TestOptWarn())
|
||||
|
||||
b.Assert(err, qt.IsNil)
|
||||
b.AssertLogContains("WARN error: KaTeX parse error: Undefined control sequence: \\foo")
|
||||
})
|
||||
}
|
||||
|
||||
func TestToMathBigAndManyExpressions(t *testing.T) {
|
||||
t.Parallel()
|
||||
|
||||
filesTemplate := `
|
||||
-- hugo.toml --
|
||||
disableKinds = ['rss','section','sitemap','taxonomy','term']
|
||||
[markup.goldmark.extensions.passthrough]
|
||||
enable = true
|
||||
[markup.goldmark.extensions.passthrough.delimiters]
|
||||
block = [['\[', '\]'], ['$$', '$$']]
|
||||
inline = [['\(', '\)'], ['$', '$']]
|
||||
-- content/p1.md --
|
||||
P1_CONTENT
|
||||
-- layouts/index.html --
|
||||
Home.
|
||||
-- layouts/_default/single.html --
|
||||
Content: {{ .Content }}|
|
||||
-- layouts/_default/_markup/render-passthrough.html --
|
||||
{{ $opts := dict "throwOnError" false "displayMode" true }}
|
||||
{{ transform.ToMath .Inner $opts }}
|
||||
`
|
||||
|
||||
t.Run("Very large file with many complex KaTeX expressions", func(t *testing.T) {
|
||||
files := strings.ReplaceAll(filesTemplate, "P1_CONTENT", "sourcefilename: testdata/large-katex.md")
|
||||
b := hugolib.Test(t, files)
|
||||
b.AssertFileContent("public/p1/index.html", `
|
||||
<span class="katex"><math
|
||||
`)
|
||||
})
|
||||
|
||||
t.Run("Large and complex expression", func(t *testing.T) {
|
||||
// This is pulled from the file above, which times out for some reason.
|
||||
largeAndComplexeExpressions := `\begin{align*} \frac{\pi^2}{6}&=\frac{4}{3}\frac{(\arcsin 1)^2}{2}\\ &=\frac{4}{3}\int_0^1\frac{\arcsin x}{\sqrt{1-x^2}}\,dx\\ &=\frac{4}{3}\int_0^1\frac{x+\sum_{n=1}^{\infty}\frac{(2n-1)!!}{(2n)!!}\frac{x^{2n+1}}{2n+1}}{\sqrt{1-x^2}}\,dx\\ &=\frac{4}{3}\int_0^1\frac{x}{\sqrt{1-x^2}}\,dx +\frac{4}{3}\sum_{n=1}^{\infty}\frac{(2n-1)!!}{(2n)!!(2n+1)}\int_0^1x^{2n}\frac{x}{\sqrt{1-x^2}}\,dx\\ &=\frac{4}{3}+\frac{4}{3}\sum_{n=1}^{\infty}\frac{(2n-1)!!}{(2n)!!(2n+1)}\left[\frac{(2n)!!}{(2n+1)!!}\right]\\ &=\frac{4}{3}\sum_{n=0}^{\infty}\frac{1}{(2n+1)^2}\\ &=\frac{4}{3}\left(\sum_{n=1}^{\infty}\frac{1}{n^2}-\frac{1}{4}\sum_{n=1}^{\infty}\frac{1}{n^2}\right)\\ &=\sum_{n=1}^{\infty}\frac{1}{n^2} \end{align*}`
|
||||
files := strings.ReplaceAll(filesTemplate, "P1_CONTENT", fmt.Sprintf(`---
|
||||
title: p1
|
||||
---
|
||||
|
||||
$$%s$$
|
||||
`, largeAndComplexeExpressions))
|
||||
|
||||
b := hugolib.Test(t, files)
|
||||
b.AssertFileContent("public/p1/index.html", `
|
||||
<span class="katex"><math
|
||||
`)
|
||||
})
|
||||
|
||||
//
|
||||
}
|
||||
|
||||
func TestToMathMacros(t *testing.T) {
|
||||
t.Parallel()
|
||||
|
||||
|
|
|
|||
Loading…
Reference in a new issue