91Ó°ÊÓ

Show that Stirling's formula $$ s ! \approx \sqrt{2 \pi s s}^{s} e^{-s} $$ holds for complex values of \(s\) (with \(\Re(s)\) large and positive). Hint. This involves assigning a phase to \(s\) and then demanding that \(\Im[s f(z)]\) \(=\) constant in the vicinity of the saddle point.

Given \(u(x)=x /\left(x^{2}+1\right)\) and \(v(x)=-1 /\left(x^{2}+1\right)\), show by direct evaluation of each integral that $$ \begin{aligned} \int_{-\infty}^{\infty}|u(x)|^{2} d x=\int_{-\infty}^{\infty}|v(x)|^{2} d x \\\ \text { ANS. } \int_{-\infty}^{\infty}|u(x)|^{2} d x=\int_{-\infty}^{\infty}|v(x)|^{2} d x=\frac{\pi}{2} . \end{aligned} $$

Show that $$ \int_{0}^{\pi} \frac{d \theta}{(a+\cos \theta)^{2}}=\frac{\pi a}{\left(a^{2}-1\right)^{3 / 2}}, \quad a>1 . $$

Show that $$ \int_{0}^{2 \pi} \frac{d \theta}{(1-2 t \cos \theta+t)^{2}}=\frac{2 \pi}{1-t^{2}}, \quad \text { for }|t|<1 . $$ What happens if \(|t|>1 ?\) What happens if \(|t|=1 ?\)

Evaluate $$ \int_{-\infty}^{\infty} \frac{\cos b x-\cos a x}{x^{2}} d x, \quad a>b>0 $$

Prove that $$ \int_{-\infty}^{\infty} \frac{\sin ^{2} x}{x^{2}} d x=\frac{\pi}{2} $$ Hint. \(\sin ^{2} x=\frac{1}{5}(1-\cos 2 x)\).

A quantum mechanical calculation of a transition probability leads to the function \(f(t, \omega)=2(1-\cos \omega t) / \omega^{2}\). Show that $$ \int_{-\infty}^{\infty} f(t, \omega) d \omega=2 \pi t . $$

Show that \((a>0)\) (a) \(\int_{-\infty}^{\infty} \frac{\cos x}{x^{2}+a^{2}} d x=\frac{\pi}{a} e^{-a}\). How is the right-hand side modified if \(\cos x\) is replaced by \(\cos k x\) ? (b) \(\int_{-\infty}^{\infty} \frac{x \sin x}{x^{2}+a^{2}} d x=\pi e^{-a}\). How is the right-hand side modified if \(\sin x\) is replaced by \(\sin k x ?\) These integrals may also be interpreted as Fourier cosine and sine transforms see Chapter 15 .

Show that $$ \int_{0}^{\infty} \frac{d x}{\left(x^{2}+a^{2}\right)^{2}}=\frac{\pi}{4 a^{3}}, \quad a>0 $$

Evaluate $$ \int_{-\infty}^{\infty} \frac{x^{2}}{1+x^{4}} d x $$