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I have seen the Fresnel integral

$$\int_0^\infty \sin x^2\, dx = \sqrt{\frac{\pi}{8}}$$

evaluated by contour integration and other complex analysis methods, and I have found these methods to be the standard way to evaluate this integral. I was wondering, however, does anyone know a real analysis method to evaluate this integral?

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See mathdl.maa.org/images/cms_upload/Chen-CMJ0926332.pdf. –  KCd Aug 29 '12 at 1:56

5 Answers 5

up vote 11 down vote accepted

Let $u=x^2$, then $$ \int_0^\infty \sin(u) \frac{\mathrm{d} u}{2 \sqrt{u}} $$ The real analysis way of evaluating this integral is to consider a parametric family: $$\begin{eqnarray} I(\epsilon) &=& \int_0^\infty \frac{\sin(u)}{2 \sqrt{u}} \mathrm{e}^{-\epsilon u} \mathrm{d} u = \frac{1}{2} \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}\int_0^\infty u^{2n+\frac{1}{2}} \mathrm{e}^{-\epsilon u} \mathrm{d} u \\ &=& \frac{1}{2} \sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!} \Gamma\left(2n+\frac{3}{2}\right) \epsilon^{-\frac{3}{2}-2n} \\ &=& \frac{1}{2 \epsilon^{3/2}} \sum_{n=0}^\infty \left(-\frac{1}{\epsilon^2}\right)^n\frac{\Gamma\left(2n+\frac{3}{2}\right)}{\Gamma\left(2n+2\right)} \\ &\stackrel{\Gamma-\text{duplication}}{=}&\frac{1}{2 \epsilon^{3/2}} \sum_{n=0}^\infty \left(-\frac{1}{\epsilon^2}\right)^n\frac{\Gamma\left(n+\frac{3}{4}\right)\Gamma\left(n+\frac{5}{4}\right)}{\sqrt{2} n! \Gamma\left(n+\frac{3}{2}\right)} \\ &=& \frac{1}{(2 \epsilon)^{3/2}} \frac{\Gamma\left(\frac{3}{4}\right)\Gamma\left(\frac{5}{4}\right)}{\Gamma\left(\frac{3}{2}\right)} {}_2F_1\left(\frac{3}{4}, \frac{5}{4}; \frac{3}{2}; -\frac{1}{\epsilon^2}\right) \\ &\stackrel{\text{Euler integral}}{=}& \frac{1}{(2 \epsilon)^{3/2}} \frac{\Gamma\left(\frac{3}{4}\right)\Gamma\left(\frac{5}{4}\right)}{\Gamma\left(\frac{3}{2}\right)} \frac{1}{\operatorname{B}\left(\frac{5}{4}, \frac{3}{2}-\frac{5}{4}\right)} \int_0^1 x^{\frac{5}{4}-1} (1-x)^{\frac{3}{2}-\frac{5}{4} -1} \left(1+\frac{x}{\epsilon^2}\right)^{-3/4} \mathrm{d} x \\ &=& \frac{1}{2^{3/2}} \frac{\Gamma\left(\frac{3}{4}\right)\Gamma\left(\frac{5}{4}\right)}{\Gamma\left(\frac{3}{2}\right)} \frac{\Gamma\left(\frac{3}{2}\right)}{\Gamma\left(\frac{5}{4}\right) \Gamma\left(\frac{1}{4}\right)} \int_0^1 x^{\frac{5}{4}-1} (1-x)^{\frac{1}{4} -1} \left(\epsilon^2+x\right)^{-3/4} \mathrm{d} x \end{eqnarray} $$ Now we are ready to compute $\lim_{\epsilon \to 0} I(\epsilon)$: $$\begin{eqnarray} \lim_{\epsilon \to 0} I(\epsilon) &=& \frac{1}{2^{3/2}} \frac{\Gamma\left(\frac{3}{4}\right)}{\Gamma\left(\frac{1}{4}\right)} \int_0^1 x^{\frac{1}{2}-1} \left(1-x\right)^{\frac{1}{4}-1} \mathrm{d} x = \frac{1}{2^{3/2}} \frac{\Gamma\left(\frac{3}{4}\right)}{\Gamma\left(\frac{1}{4}\right)} \frac{\Gamma\left(\frac{1}{2}\right) \Gamma\left(\frac{1}{4}\right)}{\Gamma\left(\frac{3}{4}\right)} \\ &=& \frac{1}{2^{3/2}} \Gamma\left(\frac{1}{2}\right) = \frac{1}{2} \sqrt{\frac{\pi}{2}} \end{eqnarray} $$

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It seems that you have assumed $\epsilon > 1$ in the intermediate steps, and identified $I(\epsilon)$ with the last line for all $\epsilon > 0$ by analytic continuation argument. Is it right? –  sos440 Aug 28 '12 at 4:28
The original integral $I(\epsilon)$ is defined for $\epsilon>0$ and is a continuous function of $\epsilon$. The convergence of the series requires $\epsilon>1$, but the Euler integral now converges for $\epsilon>0$. By the principle of analytic continuation $I(\epsilon)$ equals to the Euler integral. –  Sasha Aug 28 '12 at 4:35
Thanks, now I clearly understood (except the step using hypergeometric series, which I barely know.) –  sos440 Aug 28 '12 at 4:41

Aside from some trigonometric substitutions and identities, we will need the following identity, which can be shown using integration by parts twice: $$ \int_0^{\infty}\cos(\alpha t)e^{-\lambda t}\,\mathrm{d}t=\frac{\lambda}{\alpha^2+\lambda^2}\tag{1} $$ We will also use the standard arctangent integral: $$ \int_0^\infty\frac{\mathrm{d}t}{a^2+t^2}=\frac\pi{2a}\tag{2} $$ Now $$ \begin{align} &\left(\int_0^\infty\color{#C00000}{\sin}(x^2) e^{-\lambda x^2}\,\mathrm{d}x\right)^2\\ &=\int_0^\infty\int_0^\infty \color{#C00000}{\sin}(x^2)\color{#C00000}{\sin}(y^2) e^{-\lambda(x^2+y^2)}\,\mathrm{d}y\,\mathrm{d}x\tag{3.1}\\ &=\frac12\int_0^\infty\int_0^\infty \left(\cos(x^2-y^2) \color{#FF0000}{-}\cos(x^2+y^2)\right) e^{-\lambda(x^2+y^2)}\,\mathrm{d}y\,\mathrm{d}x\tag{3.2}\\ &=\frac12\int_0^{\pi/2}\int_0^\infty \left(\cos(r^2\cos(2\phi)) \color{#FF0000}{-}\cos(r^2)\right)e^{-\lambda r^2} \,r\,\mathrm{d}r\,\mathrm{d}\phi\tag{3.3}\\ &=\frac14\int_0^{\pi/2}\int_0^\infty \left(\cos(s\cos(2\phi)) \color{#FF0000}{-}\cos(s)\right) e^{-\lambda s} \,\mathrm{d}s\,\mathrm{d}\phi\tag{3.4}\\ &=\frac14\int_0^{\pi/2}\left(\frac{\lambda}{\cos^2(2\phi)+\lambda^2} \color{#FF0000}{-}\frac{\lambda}{1+\lambda^2}\right)\,\mathrm{d}\phi\tag{3.5}\\ &=\frac12\int_0^{\pi/4} \frac{\lambda}{\cos^2(2\phi)+\lambda^2}\,\mathrm{d}\phi \color{#FF0000}{-}\frac{\lambda\pi/8}{1+\lambda^2}\tag{3.6}\\ &=\frac14\int_0^{\pi/4} \frac{\lambda\,\mathrm{d}\tan(2\phi)} {1+\lambda^2+\lambda^2\tan^2(2\phi)} \color{#FF0000}{-}\frac{\lambda\pi/8}{1+\lambda^2}\tag{3.7}\\ &=\frac14\int_0^\infty\frac{\mathrm{d}t}{1+\lambda^2+t^2} \color{#FF0000}{-}\frac{\lambda\pi/8}{1+\lambda^2}\tag{3.8}\\ &=\frac{\pi/8}{\sqrt{1+\lambda^2}} \color{#FF0000}{-}\frac{\lambda\pi/8}{1+\lambda^2}\tag{3.9} \end{align} $$

$(3.1)$ change the square of the integral into a double integral

$(3.2)$ use $2\color{#C00000}{\sin}(A)\color{#C00000}{\sin}(B)=\cos(A-B)\color{#FF0000}{-}\cos(A+B)$

$(3.3)$ convert to polar coordinates

$(3.4)$ substitute $s=r^2$

$(3.5)$ apply $(1)$

$(3.6)$ pull out the constant and apply symmetry to reduce the domain of integration

$(3.7)$ multiply numerator and denominator by $\sec^2(2\phi)$

$(3.8)$ substitute $t=\lambda\tan(2\phi)$

$(3.9)$ apply $(2)$

Finally, take the square root of both sides of $(3)$ and let $\lambda\to0^+$ to get $$ \int_0^\infty\color{#C00000}{\sin}(x^2)\,\mathrm{d}x=\sqrt{\frac\pi8}\tag{4} $$


I just noticed that the same proof works for $$ \int_0^\infty\cos(x^2)\,\mathrm{d}x=\sqrt{\frac\pi8}\tag{5} $$ if each red $\color{#C00000}{\sin}$ is changed to $\cos$ and each red $\color{#FF0000}{-}$ sign is changed to $+$.

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@Sasha: this answer should be better than my last :-) –  robjohn Sep 3 '12 at 3:27
I love this method! It is a novel application of the polar coordinate transform for me. –  sos440 Sep 4 '12 at 6:49
This is nice that only standard "freshmen calculus" knowledge is used. –  i707107 Oct 25 '13 at 1:51
Actually, justification of $\lambda\rightarrow 0$ would be something nontrivial. –  i707107 Nov 6 '13 at 9:23
I mean in the integral, explaining why the limit of LHS is the value at 0. –  i707107 Nov 6 '13 at 9:29

First, use the substitution $t = x^2$ to obtain

$$ I := \int_{0}^{\infty}\sin\left(x^2\right)\;dx=\int_{0}^{\infty}\frac{\sin t}{2\sqrt{t}}\;dt. $$

This integral converges conditionally, thus we make an integration by parts to obtain an absolutely convergent integral as follows:

$$I = \left[\frac{1-\cos t}{2\sqrt{t}}\right]_{0}^{\infty}-\int_{0}^{\infty}\left(\frac{d}{dt}\frac{1}{2\sqrt{t}}\right)(1 - \cos t)\;dt =\int_{0}^{\infty}\frac{1 - \cos t}{4t^{3/2}}\;dt.$$

Now, from gamma integral,

$$ \begin{align*} I &=\frac{1}{4\Gamma(3/2)}\int_{0}^{\infty}\left(\frac{\Gamma(3/2)}{t^{3/2}}\right)(1 - \cos t)\;dt \\ &=\frac{1}{4\Gamma(3/2)}\int_{0}^{\infty}\left(\int_{0}^{\infty}u^{1/2}e^{-tu}\;du\right)(1 - \cos t)\;dt\\ &=\frac{1}{4\Gamma(3/2)}\int_{0}^{\infty}\int_{0}^{\infty}u^{1/2}e^{-tu}(1 - \cos t)\;dudt \\ &=\frac{1}{2\sqrt{\pi}}\int_{0}^{\infty}\int_{0}^{\infty}u^{1/2}e^{-tu}(1 - \cos t)\;dudt, \end{align*} $$

where in the last line we have used the fact that $\Gamma(3/2) = \frac{1}{2}\sqrt{\pi}$, which is a direct consequence of the Gaussian integral. (Of course, this integral can be evaluated by a famous real analysis technique.) By Tonelli's theorem we can change the order of integration, and with the substitution $u = v^2$ we obtain

$$\begin{align*}I &=\frac{1}{2\sqrt{\pi}}\int_{0}^{\infty}\int_{0}^{\infty}u^{1/2}e^{-tu}(1 - \cos t)\;dtdu \\ &= \frac{1}{2\sqrt{\pi}} \int_{0}^{\infty}\frac{u^{1/2}}{u(1+u^2)}\;du = \frac{1}{2\sqrt{\pi}} \int_{0}^{\infty} \frac{2}{1+v^4} \; dv. \end{align*}$$

To evaluate the last integral, we use the following decomposition

$$ \frac{2}{1+v^4} = \frac{1+v^{-2}}{(v-v^{-1})^2-2} - \frac{1-v^{-2}}{(v+v^{-1})^2-2}. $$

Thus with the substitution $z = v - v^{-1}$ and $w = v + v^{-1}$, the integral becmes

$$\begin{align*}I &= \frac{1}{2\sqrt{\pi}} \left( \int_{0}^{\infty} \frac{d(v-v^{-1})}{(v-v^{-1})^2+2} - \int_{0}^{\infty} \frac{d(v+v^{-1})}{(v+v^{-1})^2-2} \right) \\ &= \frac{1}{2\sqrt{\pi}} \left( \int_{-\infty}^{\infty} \frac{dz}{z^2+2} - \int_{\infty}^{\infty} \frac{dw}{w^2-2} \right) = \frac{1}{2\sqrt{\pi}} \left( \frac{\pi}{\sqrt{2}} - 0 \right) = \sqrt{\frac{\pi}{8}}. \end{align*}$$

Only a slight modification of this argument immediately yields

$$ \int_{0}^{\infty} \cos\left(x^2\right) \; dx = \sqrt{\frac{\pi}{8}}. $$

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@Sasha, Thank you. At first, I mistakenly posted a calculation for $\int_{0}^{\infty}\cos\left(x^2\right)\;dx$. When making amends in a hurry, I forgot fixing that. –  sos440 Aug 28 '12 at 4:02
This way of evaluating $\int_0^\infty \frac{dv}{1+v^4}$ is new to me. Another, maybe shorter way, would be use $v=\left(\frac{u}{1-u}\right)^{1/4}$: $$\int_0^\infty \frac{dv}{1+v^4} = \int_0^1 \frac{1}{4} (1-u)^{3/4-1} u^{1/4-1} \mathrm{d} u = \frac{1}{4} \operatorname{B}\left(\frac{1}{4}, \frac{3}{4}\right) = \frac{1}{4} \sqrt{2} \pi $$ –  Sasha Aug 28 '12 at 4:15
@Sasha, I also strongly agree that the use of beta function saves an enormous amount of effort. I also enjoy exploiting it. Here, however, I decided to try some sort of minimalism in a burden of background knowledge, to make a partial excuse for an alleged claim that real-analysis-approach is much harder. –  sos440 Aug 28 '12 at 4:24

Here is paper that addresses this exact question by H. Flanders, "On Fresnel integrals", American Mathematical Monthly, vol. 89, no. 4, 1982, pp. 264-266.

Here is another paper on the topic that does not require a subscription.

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Integrate $\displaystyle{J \equiv \int_{0}^{\infty}{\rm e}^{{\rm i}x^{2}}\,{\rm d}x}$. $\quad\Im J = ?$.

\begin{eqnarray*} J^{2} & = & \int_{0}^{\infty}{\rm e}^{{\rm i}x^{2}}\,{\rm d}x \int_{0}^{\infty}{\rm e}^{{\rm i}y^{2}}\,{\rm d}y = {\pi \over 2}\int_{0}^{\infty}{\rm e}^{{\rm i}\,\rho^{2}}\rho\,{\rm d}\rho = {\pi \over 4}\int_{0}^{\infty}{\rm e}^{{\rm i}\,\rho}\,{\rm d}\rho \\[3mm] & = & {\pi \over 4}\int_{-\infty}^{\infty} \left(% {\rm i}\int_{-\infty}^{\infty}{{\rm d}k \over 2\pi}\, {{\rm e}^{-{\rm i}k\rho} \over k + {\rm i}0^{+}} \right) {\rm e}^{{\rm i}\,\rho}\,{\rm d}\rho = {\rm i}\,{\pi \over 4}\int_{-\infty}^{\infty}{{\rm d}k \over k + {\rm i}0^{+}} \int_{-\infty}^{\infty}{{\rm d}\rho \over 2\pi}\,{\rm e}^{{\rm i}\left(1 - k\right)\rho} \\[3mm] & = & {\rm i}\,{\pi \over 4}\int_{-\infty}^{\infty}{{\rm d}k \over k + {\rm i}0^{+}} \delta\left(1 - k\right) = {\rm i}\,{\pi \over 4}{1 \over 1 + {\rm i}0^{+}} = {\rm i}\,{\pi \over 4}\left\lbrack 1 - {\rm i}\pi\,\delta\left(1\right)\right\rbrack = {\rm i}\,{\pi \over 4} = {\rm e}^{{\rm i}\pi/2}\,{\pi \over 4} \\[1cm]&&\mbox{} \end{eqnarray*}

$$ J = \sqrt{\,{\rm e}^{{\rm i}\pi/2}\,{\pi \over 4}\,} = {\rm e}^{{\rm i}\pi/4}\,\sqrt{\,\pi \over 4\,} $$

$$ \int_{0}^{\infty}\sin\left(x^{2}\right)\,{\rm d}x = \Im J = \overbrace{\quad\sin\left(\pi \over 4\right)\quad}^{=\ 1/\sqrt{\,2\,}} \sqrt{\,\pi \over 4\,} = \sqrt{\,\pi \over 8\,} $$

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