Harmonic Functions and Poisson Integral Formula

(20 pts) Prove the Cauchy Integral Formula: Let \(f\) be analytic in an open domain \(D\), and let \(z\in D\). Then for any non-negative integer \(n\), \[ f^{(n)}(z)=\frac {n!}{2\pi i}\oint _C \frac {f(w)}{(w-z)^{n+1}}\,dw, \] where \(C\) is any simple contour in \(D\) enclosing \(z\). You may take the \(n=0,1\) versions of this formula as given.

Show that \[ \int _0^\infty \frac {\cos 3x}{(x^2+1)^2}\,dx=\frac {\pi }{e^3}. \] Clearly indicate the contour of integration and the treatment of the integral along each segment.

Complete the following:

1. (5 pts) Consider two entire functions with no zeros and having a ratio equal to unity at infinity. Use Liouville’s Theorem to show that they are in fact the same function. Justify all your steps, including satisfying the assumptions of Liouville’s theorem.
2. (5 pts) Let \(f(z)\) be analytic in and on a circle \(C\) with center \(w\) and radius \(\rho >0\). From the Cauchy Integral Formula show that \[ f(w)=\frac {1}{2\pi }\int _0^{2\pi } f\!\left (w+\rho e^{i\theta }\right )\,d\theta . \]

Evaluate the following integral: \[ I=\int _0^\infty \frac {\cos x}{x^2+1}\,dx. \] Clearly indicate the contour of integration and the treatment of the integral along each segment.

Let \(C\) be the unit circle centered at the origin. Cauchy’s integral formula implies that \[ f(z)=\frac {1}{2\pi i}\oint _C \frac {f(\xi )}{\xi -z}\,d\xi ,\qquad |z|<1. \]

1. Perform the change of variables \(\xi =e^{i\theta }\) to show that \[ f(z)=\frac {1}{2\pi }\int _0^{2\pi }\frac {f(\xi )\xi }{\xi -z}\,d\theta , \] where \(z\) lies inside the unit circle. Explain why \[ 0=\frac {1}{2\pi }\int _0^{2\pi }\frac {f(\xi )\xi }{\xi -1/\bar {z}}\,d\theta . \]
2. Using the fact that \(\xi =1/\bar {\xi }\) on the unit circle, show that \[ f(z)=\frac {1}{2\pi }\int _0^{2\pi } f(\xi ) \left (\frac {\xi }{\xi -z}\pm \frac {\bar {z}}{\bar {\xi }-\bar {z}}\right )\,d\theta . \]
3. Taking the plus sign in part (b), deduce the Poisson formula for the real part \(u(r,\phi )\) of \(f(z)\) with \(z=re^{i\phi }\): \[ u(r,\phi )=\frac {1}{2\pi }\int _0^{2\pi } \frac {1-r^2}{1-2r\cos (\phi -\theta )+r^2}u(1,\theta )\,d\theta . \] This shows that the real part of \(f(z)\) on the unit circle determines the real part of \(f(z)\) everywhere inside the unit circle.

1. Use principal value integrals to show that \[ \int _0^\infty \frac {\cos kx-\cos mx}{x^2}\,dx=-\frac {\pi }{2}(k-m),\qquad k,m>0. \]
2. Deduce from (a) that \[ \int _0^\infty \frac {\sin ^2 x}{x^2}\,dx=\frac {\pi }{2}. \]

Consider the Dawson’s integral \[ D(x)=e^{-x^2}\int _0^x e^{t^2}\,dt \qquad \text {as }x\to \infty . \]

1. Find the first non-constant term of the asymptotic expansion of \(D(x)\) and show that the relationship is asymptotic.
2. Find the full asymptotic series of \(D(x)\).

HINT: Introduce a cutoff parameter.

1. Prove: If a function is analytic then its real and imaginary parts are harmonic.
2. Prove: If a function \(u(x,y)\) is harmonic in a domain \(D\), then \(u(x,y)\) cannot attain a strict local maximum in \(D\). [Hint: If \(f(z)=u+iv\), then function \(\phi (z)=e^{f(z)}\) has absolute value \(\lvert \phi (z)\rvert =e^u\).]

Find the three-term asymtotic expansion of the integral \[ I(s)=\int _0^1 \ln (t)\,e^{ist}\,dt \] with large real parameter \(s\to +\infty \).

Find the leading behavior, as \(s\to +\infty \), of the integral

1. \[ I(s)=\int _0^3 \frac {1}{\sqrt {x^2+2x}}e^{-sx}\,dx \]
2. \[ I(s)=\int _0^{\pi /2} e^{is\cos x}\,dx \]

Consider integral \[ I(s)=\int _C \frac {e^{s(z^2-1)}}{z-1}\,dz \] with large parameter \(s\to +\infty \); \(C\) is the vertical line \(\operatorname {Im}z=3\), from \(z=3-i\infty \) to \(z=3+i\infty \).

1. What is the saddle point \(z_0\)?

   What is the path \(C_0\) of steepest descent from \(z_0\)?

   What is the path of steepest ascent from \(z_0\)?

2. Find the three-term asymtotic expansion of the integral.

Evaluate the integral \[ \int _0^\infty e^{ix^2}\,dx. \]

Find the leading behavior, as \(s\to +\infty \), of the integral

1. \[ I=\int _0^3\frac {1}{\sqrt {x^2+2x}}e^{-sx}\,dx, \]
2. \[ I=\int _0^{\pi /2} e^{is\cos x}\,dx. \]

Calculate the integral \[ I=\int _0^\infty \frac {x^\alpha }{1+x}\,dx, \] where \(\alpha \) is a real number for which the integral converges.

Let \[ f(x)= \begin {cases} \frac {100}{n}, & x\ \text {rational},\ x=\frac {m}{n},\\ -2, & x\ \text {irrational}. \end {cases} \] Does the Riemann integral of \(f(x)\) over the interval \([-1,1]\) exist? If so, compute it.

Does the Lebesgue integral of \(f(x)\) over the interval \([-1,1]\) exist? If so, compute it.

Why is the Lebesgue integral used in defining the Hilbert space \(L^2[0,1]\), and not the Riemann integral?

Evaluate the integral \[ I=\int _0^\infty \frac {x^\alpha \,dx}{(x^2+1)} \] (where \(\alpha \) is a parameter, such that the integral converges).

Evaluate the integral \[ I=\int _0^\infty \frac {\cos ax}{(x+1)^2}\,dx. \]

Formulate and prove the Cauchy integral formula.

Evaluate the integral \[ I=\int _0^\infty \frac {x^\alpha \,dx}{(x+1)^2} \] (where \(\alpha \) is a constant parameter, so that the integral converges).

Evaluate the integral \[ I=\int _0^\infty \frac {x^\alpha \,dx}{(x-1)^2+1}. \] (where \(\alpha \) is a constant, such that the integral converges: \(|\alpha |<1\)).

Evaluate the integral \[ I=\int _0^\infty \frac {x^\alpha }{x^2+1}\,dx \] where \(\alpha \) is a parameter, \(-1<\alpha <1\).