Principal Value Integrals

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 \[ f(w)=\frac {1}{2\pi }\int _0^{2\pi } f\!\left (w+\rho e^{i\theta }\right )\,d\theta . \]

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 \[ \int _0^\infty \frac {\sin ^2 x}{x^2}\,dx=\frac {\pi }{2}. \]

Use the method of steepest descent to estimate the integral \[ I(s)=\int _0^1 e^{s(-z^3+iz^3)}\,dz \] for large positive real values of \(s\).

Formulate and prove the Cauchy integral formula.

Suppose that \(\psi _+(z)\) is analytic on the upper half plane and \(\psi _-(z)\) is analytic on the lower half plane such that on the real axis \[ \psi _+(x)-\psi _-(x)=f(x) \] Assuming that the integrable function \(f(x)\) can be analytically extended in a neighborhood of the real axis, show that a solution to this problem is given by the Cauchy integral \[ \Psi (z)=\frac {1}{2\pi i}\int _{-\infty }^{\infty }\frac {f(\xi )}{\xi -z}\,d\xi \] with \(\psi _\pm (x)=\lim _{\epsilon \to 0+}\Psi (x\pm i\epsilon )\). Hence, establish that \[ \psi _+(x)+\psi _-(x)=\frac {1}{\pi i}\operatorname {PV} \int _{-\infty }^{\infty }\frac {f(\xi )}{\xi -x}\,d\xi \] where \(\operatorname {PV}\) denotes the principal value.