From e1e1baa1bd33181b32c58524bc21ef21da4880db Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Bj=C3=B8rn=20Erik=20Pedersen?= Date: Sun, 11 Aug 2024 20:31:17 +0200 Subject: [PATCH] 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 --- common/types/types.go | 24 ++ internal/warpc/js/common.js | 12 +- internal/warpc/js/greet.bundle.js | 4 +- internal/warpc/js/renderkatex.bundle.js | 18 +- internal/warpc/js/renderkatex.js | 9 +- internal/warpc/katex.go | 1 - internal/warpc/warpc.go | 5 +- internal/warpc/wasm/greet.wasm | Bin 1979 -> 2226 bytes internal/warpc/wasm/renderkatex.wasm | Bin 472126 -> 472553 bytes .../go_templates/htmltemplate/content.go | 3 +- .../htmltemplate/hugo_template.go | 12 + tpl/transform/testdata/large-katex.md | 373 ++++++++++++++++++ tpl/transform/transform.go | 27 +- tpl/transform/transform_integration_test.go | 105 +++++ 14 files changed, 569 insertions(+), 24 deletions(-) create mode 100644 tpl/transform/testdata/large-katex.md diff --git a/common/types/types.go b/common/types/types.go index e3850ddac..322dfe592 100644 --- a/common/types/types.go +++ b/common/types/types.go @@ -120,3 +120,27 @@ var InvocationCounter atomic.Int64 func NewBool(b bool) *bool { return &b } + +// PrintableValueProvider is implemented by types that can provide a printable value. +type PrintableValueProvider interface { + PrintableValue() any +} + +var _ PrintableValueProvider = Result[any]{} + +// Result is a generic result type. +type Result[T any] struct { + // The result value. + Value T + + // The error value. + Err error +} + +// PrintableValue returns the value or panics if there is an error. +func (r Result[T]) PrintableValue() any { + if r.Err != nil { + panic(r.Err) + } + return r.Value +} diff --git a/internal/warpc/js/common.js b/internal/warpc/js/common.js index ba80e06f9..90ea1a1bc 100644 --- a/internal/warpc/js/common.js +++ b/internal/warpc/js/common.js @@ -30,13 +30,23 @@ export function readInput(handle) { currentLine = [...currentLine, ...buffer.subarray(0, bytesRead)]; + // Check for newline. If not, we need to read more data. + if (!currentLine.includes(10)) { + continue; + } + // Split array into chunks by newline. let i = 0; for (let j = 0; i < currentLine.length; i++) { if (currentLine[i] === 10) { const chunk = currentLine.splice(j, i + 1); const arr = new Uint8Array(chunk); - const json = JSON.parse(new TextDecoder().decode(arr)); + let json; + try { + json = JSON.parse(new TextDecoder().decode(arr)); + } catch (e) { + throw new Error(`Error parsing JSON '${new TextDecoder().decode(arr)}' from stdin: ${e.message}`); + } handle(json); j = i + 1; } diff --git a/internal/warpc/js/greet.bundle.js b/internal/warpc/js/greet.bundle.js index c1b9c5264..f6776cf53 100644 --- a/internal/warpc/js/greet.bundle.js +++ b/internal/warpc/js/greet.bundle.js @@ -1,2 +1,2 @@ -(()=>{function i(r){let e=[],a=new Uint8Array(1024);for(;;){let n=0;try{n=Javy.IO.readSync(0,a)}catch(o){if(o.message.includes("os error 29"))break;throw new Error("Error reading from stdin")}if(n<0)throw new Error("Error reading from stdin");if(n===0)break;e=[...e,...a.subarray(0,n)];let t=0;for(let o=0;t{function s(r){let e=[],c=new Uint8Array(1024);for(;;){let n=0;try{n=Javy.IO.readSync(0,c)}catch(o){if(o.message.includes("os error 29"))break;throw new Error("Error reading from stdin")}if(n<0)throw new Error("Error reading from stdin");if(n===0)break;if(e=[...e,...c.subarray(0,n)],!e.includes(10))continue;let t=0;for(let o=0;t{function Vt(r){let t=[],a=new Uint8Array(1024);for(;;){let s=0;try{s=Javy.IO.readSync(0,a)}catch(h){if(h.message.includes("os error 29"))break;throw new Error("Error reading from stdin")}if(s<0)throw new Error("Error reading from stdin");if(s===0)break;t=[...t,...a.subarray(0,s)];let o=0;for(let h=0;o15?p="\u2026"+h.slice(n-15,n):p=h.slice(0,n);var g;s+15":">","<":"<",'"':""","'":"'"},za=/[&><"']/g;function Aa(r){return String(r).replace(za,e=>Ma[e])}var wr=function r(e){return e.type==="ordgroup"||e.type==="color"?e.body.length===1?r(e.body[0]):e:e.type==="font"?r(e.body):e},Ta=function(e){var t=wr(e);return t.type==="mathord"||t.type==="textord"||t.type==="atom"},Ba=function(e){if(!e)throw new Error("Expected non-null, but got "+String(e));return e},Da=function(e){var t=/^[\x00-\x20]*([^\\/#?]*?)(:|�*58|�*3a|&colon)/i.exec(e);return t?t[2]!==":"||!/^[a-zA-Z][a-zA-Z0-9+\-.]*$/.test(t[1])?null:t[1].toLowerCase():"_relative"},N={contains:xa,deflt:wa,escape:Aa,hyphenate:Sa,getBaseElem:wr,isCharacterBox:Ta,protocolFromUrl:Da},qe={displayMode:{type:"boolean",description:"Render math in display mode, which puts the math in display style (so \\int and \\sum are large, for example), and centers the math on the page on its own line.",cli:"-d, --display-mode"},output:{type:{enum:["htmlAndMathml","html","mathml"]},description:"Determines the markup language of the output.",cli:"-F, --format "},leqno:{type:"boolean",description:"Render display math in leqno style (left-justified tags)."},fleqn:{type:"boolean",description:"Render display math flush left."},throwOnError:{type:"boolean",default:!0,cli:"-t, --no-throw-on-error",cliDescription:"Render errors (in the color given by --error-color) instead of throwing a ParseError exception when encountering an error."},errorColor:{type:"string",default:"#cc0000",cli:"-c, --error-color ",cliDescription:"A color string given in the format 'rgb' or 'rrggbb' (no #). 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r{constructor(e,t){this.mode=void 0,this.gullet=void 0,this.settings=void 0,this.leftrightDepth=void 0,this.nextToken=void 0,this.mode="math",this.gullet=new xt(e,t,this.mode),this.settings=t,this.leftrightDepth=0}expect(e,t){if(t===void 0&&(t=!0),this.fetch().text!==e)throw new M("Expected '"+e+"', got '"+this.fetch().text+"'",this.fetch());t&&this.consume()}consume(){this.nextToken=null}fetch(){return this.nextToken==null&&(this.nextToken=this.gullet.expandNextToken()),this.nextToken}switchMode(e){this.mode=e,this.gullet.switchMode(e)}parse(){this.settings.globalGroup||this.gullet.beginGroup(),this.settings.colorIsTextColor&&this.gullet.macros.set("\\color","\\textcolor");try{var e=this.parseExpression(!1);return this.expect("EOF"),this.settings.globalGroup||this.gullet.endGroup(),e}finally{this.gullet.endGroups()}}subparse(e){var t=this.nextToken;this.consume(),this.gullet.pushToken(new p0("}")),this.gullet.pushTokens(e);var a=this.parseExpression(!1);return this.expect("}"),this.nextToken=t,a}parseExpression(e,t){for(var a=[];;){this.mode==="math"&&this.consumeSpaces();var n=this.fetch();if(r.endOfExpression.indexOf(n.text)!==-1||t&&n.text===t||e&&L0[n.text]&&L0[n.text].infix)break;var s=this.parseAtom(t);if(s){if(s.type==="internal")continue}else break;a.push(s)}return this.mode==="text"&&this.formLigatures(a),this.handleInfixNodes(a)}handleInfixNodes(e){for(var t=-1,a,n=0;n=0&&this.settings.reportNonstrict("unicodeTextInMathMode",'Latin-1/Unicode text character "'+t[0]+'" used in math mode',e);var h=$[this.mode][t].group,c=m0.range(e),p;if(e1.hasOwnProperty(h)){var g=h;p={type:"atom",mode:this.mode,family:g,loc:c,text:t}}else p={type:h,mode:this.mode,loc:c,text:t};o=p}else if(t.charCodeAt(0)>=128)this.settings.strict&&(kr(t.charCodeAt(0))?this.mode==="math"&&this.settings.reportNonstrict("unicodeTextInMathMode",'Unicode text character "'+t[0]+'" used in math mode',e):this.settings.reportNonstrict("unknownSymbol",'Unrecognized Unicode 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+$$\frac{\sin x}{x} = 1 - \frac{x^2}{3!}+\frac{x^4}{5!}-\frac{x^6}{7!}+\cdots\quad.$$ +Now substitute $ x = \sqrt{y}\ $ to arrive at +$$\frac{\sin \sqrt{y}\ }{\sqrt{y}\ } = 1 - \frac{y}{3!}+\frac{y^2}{5!}-\frac{y^3}{7!}+\cdots\quad.$$ +if we find the roots of $\frac{\sin \sqrt{y}\ }{\sqrt{y}\ } = 0 $ we find that +$ y = n^2\pi^2\ $ for $ n \neq 0 $ and $ n $ in the integers +With all of this in mind, recall that for a polynomial +$ 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} $ +$$\frac{1}{r_{1}} + \frac{1}{r_{2}} + \cdots + \frac{1}{r_{n}} = -\frac{a_{1}}{a_{0}}$$ +Treating the above series for $ \frac{\sin \sqrt{y}\ }{\sqrt{y}\ } $ as polynomial we see that +$$\frac{1}{1^2\pi^2} + \frac{1}{2^2\pi^2} + \frac{1}{3^2\pi^2} + \cdots = -\frac{-\frac{1}{3!}}{1}$$ +then multiplying both sides by $ \pi^2 $ gives the desired series. +$$\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \cdots = \frac{\pi^2}{6}$$ + +Nameless +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 $27$) +Consider +$$\begin{align} \int_{0}^{1} \int_{0}^{1} \frac{1}{1 - xy} dy dx &= \int_{0}^{1} \int_{0}^{1} \sum_{n \geq 0} (xy)^n dy dx \\ &= \sum_{n \geq 0} \int_{0}^{1} \int_{0}^{1} x^n y^n dy dx \\ &= \sum_{n \geq 1} \frac{1}{n^2} \\ \end{align}$$ +You can verify that the left hand side is indeed $\frac{\pi^2}{6}$ by letting $x = u - v$ and $y = v + u.$ + +Qiaochu Yuan +I have two favorite proofs. One is the last proof in Robin Chapman's collection; you really should take a look at it. +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. +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, +$$\frac{x}{e^x - 1} = \sum_{n \in \mathbb{Z}} \frac{2\pi i n}{x - 2 \pi i n} = \sum_{n \in \mathbb{Z}} - \left( \frac{1}{1 - \frac{x}{2\pi i n}} \right).$$ +Now we can expand each of the terms on the RHS as a geometric series, again a la Euler, to obtain +$$\frac{x}{e^x - 1} = - \sum_{n \in \mathbb{Z}} \sum_{k \ge 0} \left( \frac{x}{2\pi i n} \right)^k = \sum_{k \ge 0} (-1)^{n+1} \frac{2 \zeta(2n)}{(2\pi )^{2n}} x^{2n}$$ +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 +$$B_{2n} = (-1)^{n+1} \frac{2 \zeta(2n)}{(2\pi)^{2n}}$$ +or +$$\zeta(2n) = (-1)^{n+1} \frac{B_{2n} (2\pi)^{2n}}{2}$$ +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 = \sum_n \frac{1}{\lambda_n} $$and $$\sum_n = \int_0^1 \sum_n f_n(x) \,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. +` \ No newline at end of file diff --git a/tpl/transform/transform.go b/tpl/transform/transform.go index 293d01a00..843351702 100644 --- a/tpl/transform/transform.go +++ b/tpl/transform/transform.go @@ -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. diff --git a/tpl/transform/transform_integration_test.go b/tpl/transform/transform_integration_test.go index 9e5221c9a..f7f3f8668 100644 --- a/tpl/transform/transform_integration_test.go +++ b/tpl/transform/transform_integration_test.go @@ -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", ` +