How Can You Prove This Complex Integral Equals $\frac{\sqrt{6\pi}}{4}$?

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    2015
In summary, POTW stands for "Problem of the Week" and is a challenging problem or puzzle presented for scientific and mathematical communities to solve. The purpose of POTW #169 is to stimulate critical thinking and problem-solving skills. The difficulty level varies, but the goal is to engage in the problem-solving process and learn from it. There are no specific rules or guidelines, but a systematic and logical approach is recommended. There is no official prize, but the satisfaction and learning experience are the main rewards.
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Euge
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Here is this week's POTW:

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Prove that

$$\int_0^\infty t^{-1/2}e^{-t}\cos(t\sqrt{3})\, dt = \frac{\sqrt{6\pi}}{4}.$$
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Congratulations to Ackbach for solving this problem correctly! His solution is clever. You can read it below.
We have:
\begin{align*}
\int_0^{\infty}\frac{e^{-t}\cos(t\sqrt{3})}{\sqrt{t}} \, dt&=\text{Re}\int_0^{\infty}\frac{e^{-t}e^{it\sqrt{3}}}{\sqrt{t}} \, dt \\
&=\text{Re}\int_0^{\infty}\frac{e^{t(-1+i\sqrt{3})}}{\sqrt{t}} \, dt \qquad \left[u=\sqrt{t},\quad 2 \, du=\frac{dt}{\sqrt{t}}\right] \\
&=2\,\text{Re}\int_0^{\infty}e^{u^2(-1+i\sqrt{3})} \, du.
\end{align*}
Let $\displaystyle I=\int_0^{\infty}e^{u^2(-1+i\sqrt{3})} \, du$. Then
\begin{align*}
I^2&=\int_0^{\infty}e^{u^2(-1+i\sqrt{3})} \, du \cdot \int_0^{\infty}e^{v^2(-1+i\sqrt{3})} \, dv \\
&=\int_0^{\infty} \int_0^{\infty}e^{(-1+i\sqrt{3})(u^2+v^2)} \, du \, dv \\
&=\int_0^{\pi/2}\int_0^{\infty}e^{(-1+i\sqrt{3})r^2} \, r \, dr \, d\theta \qquad (\text{switching to polar})\\
&=\frac{\pi}{2}\int_0^{\infty}e^{(-1+i\sqrt{3})r^2} \, r \, dr \qquad \left[ s=r^2, \quad \frac{ds}{2}=r \, dr \right] \\
&=\frac{\pi}{4}\int_0^{\infty}e^{(-1+i\sqrt{3})s} \, ds \\
&=\frac{\pi}{4}\left(\frac{e^{(-1+i\sqrt{3})s}}{-1+i\sqrt{3}}\right)\Bigg|_{0}^{\infty \, \to \, 0} \\
&=\frac{\pi}{4} \, \frac{1+i\sqrt{3}}{4} \\
I^2&=\frac{\pi(1+i\sqrt{3})}{16} \\
I&=\pm\frac{\sqrt{\pi}}{2\sqrt{2}} \, e^{i\pi/6}.
\end{align*}
Now then,
\begin{align*}
\int_0^{\infty}\frac{e^{-t}\cos(t\sqrt{3})}{\sqrt{t}} \, dt&=2\,\text{Re}\int_0^{\infty}e^{u^2(-1+i\sqrt{3})} \, du \\
&=2 \, \text{Re} \left[ \pm\frac{\sqrt{\pi}}{2\sqrt{2}} \, e^{i\pi/6} \right] \\
&=\pm\frac{\sqrt{\pi}}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2} \\
&=\pm\frac{\sqrt{6\pi}}{4}.
\end{align*}
Now here is the justification for ruling out the minus sign. The function $1/(e^t \sqrt{t})$ is monotone decreasing on the interval of integration. $\cos(t\sqrt{3})$ is oscillating. Let us examine the first period of this function carefully - this would be the interval $\displaystyle (0,2\pi/\sqrt{3})$. On $(0,\pi/(2\sqrt{3}))$, the cosine function is positive, and on $(\pi/(2\sqrt{3}),\pi/\sqrt{3})$, the cosine is negative. Because the multiplying function is decreasing, the positive portion will outweigh the negative. Next, we compare the interval $(\pi/\sqrt{3},3\pi/(2\sqrt{3}))$ to $(3\pi/(2\sqrt{3}),2\pi/\sqrt{3})$. The cosine function will be negative for the former, and positive for the latter. Hence, the integral, considered on these two intervals, will be negative overall. So, on the first half-period, the integral is positive, and on the second half-period, it is negative. Because the multiplying function is decreasing, however, the positive portion will dominate the negative, making the entire integral positive. We can repeat this analysis for any period, hence we can rule out the negative sign.
 

FAQ: How Can You Prove This Complex Integral Equals $\frac{\sqrt{6\pi}}{4}$?

What does POTW stand for?

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What is the purpose of POTW #169?

The purpose of POTW #169 is to provide a stimulating and thought-provoking problem for scientists and mathematicians to solve. It encourages critical thinking and problem-solving skills, and allows individuals to showcase their knowledge and abilities.

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The difficulty level of POTW #169 varies depending on an individual's background and expertise. It may be considered challenging for some and relatively easy for others. However, the goal is not to solve the problem quickly or easily, but to engage in the problem-solving process and learn from it.

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