Cauchy Integral Formula and Integral Theorems

(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.

Complete the following:

1. (10 pts) Prove Rouche’s theorem: Assume \(f\) and \(g\) are analytic functions inside and on a simple closed contour \(C\), and suppose \(|f(z)|>|g(z)|\) for all \(z\in C\). Then \(f\) and \(f+g\) have the same number of zeros inside \(C\).
2. (10 pts) Prove the following variant of the Argument Principle: Let \(C\) be a simple closed curve with counterclockwise orientation, and suppose \(f\) is analytic inside and on \(C\), except at \(M\) poles inside \(C\) (poles are counted with weight corresponding to their multiplicity). Let \(a\in \mathbb {C}\), and assume \(f(z)+a\) has no zeros on \(C\) and has \(N_a\) zeros inside \(C\) (again, counting according to multiplicity). Without citing the Argument Principle, prove that \[ \frac {1}{2\pi i}\oint _C \frac {f'(z)}{f(z)+a}\,dz=N_a-M. \]

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.

Prove the argument principle: Assume that \(f\) is a meromorphic function inside and on a simple closed contour \(C\) (oriented in the typical counter-clockwise fashion), that \(f\) has exactly \(N\) zeros (counting multiplicity) in this region all lying within \(C\), and that \(f\) has exactly \(M\) poles (counting order), all lying within \(C\). Prove that \[ \frac {1}{2\pi i}\oint _C \frac {f'(z)}{f(z)}\,dz=N-M. \]

Evaluate the following integral by residue integration: \[ \int _0^\infty \frac {dx}{x^3+a^3},\qquad a>0. \] 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 nonlinear integral equation \[ u(x)-\lambda \int _a^b w(x,y)\sin (u(y))\,dy=\phi (x), \] where \(\lambda \in \mathbb {R}\), \(\phi \) is continuous on the interval \([a,b]\) and \(w(x,y)\) is continuous on \([a,b]\times [a,b]\). Find a constant \(C>0\) such that \(\lvert \lambda \rvert <C\) implies that the nonlinear integral equation has a unique solution \(u\).

Hint: You may assume the inequality \(\lvert \sin (x)-\sin (y)\rvert \le \lvert x-y\rvert \), for any \(x,y\in \mathbb {R}\).

Evaluate the integrals \[ \begin {array}{rcll} \text {(a)}&\displaystyle \int _{-\infty }^{\infty }\frac {\sin ax}{x}\,dx &&\text {(\(a\) is a real parameter)},\\[2.2ex] \text {(b)}&\displaystyle \int _0^\infty \frac {x^\alpha }{x+1}\,dx &&\text {(\(\alpha \) is a real parameter, \(-1<\alpha <0\))}. \end {array} \]

Consider integral \[ I(s)=\int _C \frac {e^{s z^2}}{z^2-1}\,dz, \] \(C\) is the vertical line \(\operatorname {Re}z=2\), from \(z=2-i\infty \) to \(z=2+i\infty \), with large positive parameter \(s\). Find the three-term asymptotic expansion (approximation with three non-zero terms) of this integral as \(s\to +\infty \).

Consider the nonlinear integral equation \[ f(x)-\lambda \int _a^b k(x,y;f(y))\,dy=\phi (x) \] where \(\lambda \in \mathbb {R}\), \(\phi \) is continuous on \([a,b]\), \(k\) is continuous on \(C=[a,b]\times [a,b]\times \mathbb {R}\), and \(k\) satisfies a Lipschitz condition of the form \[ \lvert k(x,y;z_1)-k(x,y;z_2)\rvert \le L\lvert z_1-z_2\rvert \] in its ‘functional argument’. Show that the nonlinear integral equation has a unique solution, \(f\), for any \(\lambda \) such that \[ \lvert \lambda \rvert <\frac {1}{L(b-a)}. \]

Calculate the following integrals (\(a\) and \(b\) are positive parameters) \[ \begin {array}{rcl} \text {(a)}\quad I(a)&=&\displaystyle \int _{-\infty }^{\infty }\frac {\sin ax}{x}\,dx,\\[2.2ex] \text {(b)}\quad I(a,b)&=&\displaystyle \int _{-\infty }^{\infty } \frac {\cos ax-\cos bx}{x^2}\,dx. \end {array} \]

Consider integral \[ I(s)=\int _C \frac {e^{s z^2}}{z^2-1}\,dz, \] \(C\) is the vertical line \(\operatorname {Re} z=3\), from \(z=3-i\infty \) to \(z=3+i\infty \), with large (real) parameter \(s\). Find the three-term asymtotic expansion (approximation with three non-zero terms) of this integral as \(s\to +\infty \).

Evaluate the integral \[ \int _0^\infty \frac {\sin (ax)}{x}\,dx \qquad \text {for }a>0. \] You must clearly show the contours of integration and treat integration over each contour separately.

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.

Evaluate the integrals

1. \[ \int _{-\infty }^{\infty }\frac {\sin x}{x}\,dx, \]
2. \[ \int _0^\infty \frac {x^\alpha }{x+1}\,dx. \]

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

Let \(D_1\) and \(D_2\) be two disjoint domains, whose boundaries share a common curve \(\Gamma \). Let

• \(f(z)\) be analytic in \(D_1\) and continuous in \(D_1\cup \Gamma \)
• \(g(z)\) be analytic in \(D_2\) and continuous in \(D_2\cup \Gamma \)
• \(f(z)=g(z)\) for \(z\in \Gamma \)

Show that the function \[ H(z)= \begin {cases} f(z), & z\in D_1,\\ f(z)=g(z), & z\in \Gamma ,\\ g(z), & z\in D_2 \end {cases} \] is analytic in \(D=D_1\cup \Gamma \cup D_2\).

[Hint: Use the Morera theorem.]

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. \[ I(s)=\int _0^{\pi /2} e^{is\cos x}\,dx \]

Evaluate the integrals:

1. \[ I=\int _0^\infty \frac {\sin x}{x}\,dx, \]
2. \[ I=\int _0^\infty \frac {x^\alpha }{2+x}\,dx \qquad (-1<\alpha <0). \]

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 of the integral \[ I=\int _0^3\frac {\cos x}{\sqrt {x}}e^{-sx}\,dx \] for large positive \(s\).

Formulate and prove Morera’s theorem.

Let \(f(x)=|x|-1\) on the interval \([-1,1]\).

1. Treating \(f\) as a distribution, calculate \[ \int _{-1}^{1} f'(x)f'(x)\,dx \quad \text {and}\quad \int _{-1}^{1} f(x)f''(x)\,dx. \]
2. Explain why \[ \int _{-1}^{1} f'(x)f''(x)\,dx \quad \text {and}\quad \int _{-1}^{1} f(x)f'''(x)\,dx \] do not make sense.
3. In general, what property is required of \(g(x)\) so that \[ \int _{-1}^{1} g(x)f^{(n)}(x)\,dx \] is well-defined for a given \(n\)th derivative?

Evaluate the integrals:

1. \[ I=\int _0^\infty \frac {\sin x}{x}\,dx, \]
2. \[ I=\int _0^\infty \frac {x^\alpha }{2+x}\,dx \qquad (-1<\alpha <1). \]

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. \[ I=\int _0^{\pi /2} e^{is\cos x}\,dx. \]

1. Find all conditions on \(\alpha ,\beta ,\gamma \), and \(f(x)\) for which solutions of \[ u''+\alpha ^2u=f(x),\qquad u(0)=\beta ,\qquad \alpha u'(0)-u'(1)=\gamma \tag {1} \] exist.
2. Find an integral representation for the solution in the case that \(\alpha =0\).

Consider the integral equation \[ \phi (x)-\lambda \int _0^\pi \sin (x+t)\phi (t)\,dt=\cos (x)+\sin (x), \qquad 0\le x\le \pi . \]

1. Find the unique solution when \(\lambda \ne \pm 2/\pi \). Why is a unique solution guaranteed to exist?
2. Show that there is no solution when \(\lambda =2/\pi \).
3. Show that when \(\lambda =-2/\pi \) there is a one-parameter family of solutions of the form \[ \phi (x)=f_p(x)+\alpha f_h(x). \] Determine \(f_h(x)\).

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

Find at least three terms of the asymptotic expansion of the integral \[ I(s)=\int _0^1 \ln t\, e^{ist}\,dt, \qquad s\text { is real and }s\to +\infty . \]

For a function \(f(z)\) which is analytic within a simple, nonintersecting, closed curve \(C\) a corollary of the Cauchy integral formula implies that for any point \(z\) inside \(C\), \[ f'(z)=\frac {1}{2\pi i}\int _C\frac {f(\xi )}{(\xi -z)^2}\,d\xi . \] Use this to prove Liouville’s theorem that any function which is bounded and analytic in the entire complex plane must be a constant, and hence prove the fundamental theorem of algebra that an \(n\)th order polynomial \(P_n(z)\) has \(n\) roots (counting multiplicity) in the complex plane.

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 \(s\).

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?

Using Green’s functions, solve the problem \[ \frac {d^2u}{dx^2}=\frac {1}{1+x^2},\qquad u(-1)=u(1)=0 \] for \(u(x)\), \(x\in [-1,1]\) (obtain an integral representation for the solution, but do not evaluate the integral).

Use Jordan’s Lemma (and describe how Jordan’s Lemma is used) to evaluate the integral \[ I=\int _0^\infty \frac {\cos ax}{x^2+1}\,dx. \]

Find the leading order term of the asymptotic expansion of the integral \[ I(s)=\int _{-\infty }^{\infty } e^{is\left (t+\frac {t^3}{3}\right )}\,dt, \qquad s\text { is real and }s\to +\infty , \] as well as a rigorous estimate of the error.

Evaluate the integral \[ I=\int _{-\infty }^{+\infty } e^{iax^2}\,dx \] (\(a\) is a positive parameter).

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

On the interval \([0,1]\) use eigenfunctions to find the resolvent of the integral equation \[ u(x)-\lambda \int _0^1 x^2y^2u(y)\,dy=f(x). \] When does a solution not exist? Write down the pseudoresolvent in this case.

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

Solve the integral equation \[ \int _{-\infty }^{\infty } k(x-y)u(y)\,dy-u(x)=f(x) \] with \(k(x)=H(x)\), the Heaviside function.

Find the leading term of the asymptotic expansion of the integral \[ I(s)=\int _{-\infty }^{\infty } e^{is\left (t+\frac {t^3}{3}\right )}\,dt, \qquad s\text { is real and }s\to +\infty , \] as well as a rigorous estimate of the error.

Consider the integral equation \[ \phi (x)-\lambda \int _0^\pi \sin (x+t)\phi (t)\,dt = \cos (x)+\sin (x), \qquad 0\le x\le \pi . \]

1. Find the unique solution when \(\lambda \ne \pm 2/\pi \).
2. Use the Fredholm alternative theorem to show that there is no solution when \(\lambda =2/\pi \).
3. Use the Fredholm alternative theorem to show that when \(\lambda =-2/\pi \) there is a one-parameter family of solutions of the form \[ \phi (x)=\frac {\cos (x)+\sin (x)}{2}+\alpha (\cos (x)-\sin (x)). \]

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 \[ \int _{-\infty }^{\infty }\frac {\sin \alpha x}{x}\,dx \] (\(\alpha \) is a constant parameter). Explain your steps.

Find the two-term asymptotic expansion of the integral \[ I(s)=\int _{-\infty }^{\infty }\frac {e^{s(ix+1)^2}}{(x+i)^2}\,dx, \qquad s\text { is real and }s\to +\infty . \]

1. Find all conditions on \(\alpha \), \(\beta \), \(\gamma \), and \(f(x)\) for which solutions of \[ u''+\alpha ^2u=f(x),\qquad u(0)=\beta ,\qquad \alpha u'(0)-u'(1)=\gamma \] exist.
2. Find an integral representation for the solution in the case that \(\alpha =0\).

Evaluate the integral \[ \int _0^\infty \frac {\sin x}{x}\,dx. \] Explain your reasoning.

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: \(|\alpha |<1\)).

Find the two-term asymptotic expansion of the integral \[ I(s)=\int _0^3 \frac {e^{s(-t^2+4t-3)}}{5+t}\,dt, \] \(s\) is real, and \(s\to +\infty \).

Prove that the nonlinear integral equation \[ u(x)-\int _0^1 \frac {u^2(y)}{4+x^2-y}\,dy=\frac {x^3}{3},\qquad 0\le x\le 1, \] has exactly one continuous solution \(u\) satisfying \(\max _{0\le x\le 1}|u(x)|\le 1\). Describe, but do not carry out, an iterative procedure for obtaining an approximation to this solution.

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.

Obtain the inverse Laplace transform of the function \[ \widehat {F}(s)=s^{-a},\qquad 0<a<1 \] by evaluating the associated Bromwich contour integral with a branch cut along the negative real axis. Express the answer in terms of the gamma function \[ \Gamma (z)=\int _0^\infty t^{z-1}e^{-t}\,dt. \]

1. Find all conditions on \(\alpha \), \(\beta \), \(\gamma \), and \(f(x)\) for which solutions of \[ u''+\alpha ^2u=f(x),\qquad u(0)=\beta ,\qquad \alpha u'(0)-u'(1)=\gamma \] exist.
2. Find an integral representation for the solution in the case that \(\alpha =0\).

Find an integral representation for the solution of the telegrapher’s equation \[ u_{tt}+au_t+bu=u_{xx} \] with \(u(0,t)=f(t)\) and \(u_x(0,t)=g(t)\) specified for all time.

Suppose \(\Omega \) is some closed, bounded domain in two spatial dimensions.

1. Under what conditions does the equation \(\nabla ^2\phi =0\) in \(\Omega \), \(\mathbf {n}\cdot \nabla \phi =f\) on \(\partial \Omega \), have a solution? Is it unique?
2. Find an integral equation for \(\phi \) on the boundary \(\partial \Omega \). Be as explicit as possible.
3. Under what conditions does this integral equation have a solution?

Evaluate the integral \[ I=\int _0^\infty \frac {\sin x}{x}\,dx. \]

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

Consider the integral equation \[ u(x)=f(x)+\frac {1}{2}\int _0^1 xtu(t)\,dt,\qquad \text {for }0\le x\le 1 \] where \(f(x)\) is a prescribed function.

1. Show that the equation has at most one solution.
2. Derive a formula for the solution.
3. Calculate the solution for the special case \(f(x)=5x/6\).

Using complex integration, evaluate the integral \[ I=\int _0^{2\pi }\frac {1}{1+\frac {1}{3}\cos \phi }\,d\phi . \]

Evaluate the integral \[ I=\int _0^\infty \frac {x^2\cos x}{x^4+a^4}\,dx \] by contour integration. Use the Jordan lemma.

Using an integral representation for the \(n\)-th order Bessel function \(J_n(x)\), find the leading order asymptotic representation of \(J_n(x)\) for large positive \(x\).