Real Integral Evaluation by Complex Methods

Use residue calculus to show that \[ \int _0^\infty \frac {\log x}{x^3-1}\,dx=\frac {4\pi ^2}{27}. \] Hint: Use an appropriate branch of the function \((\log z)/(z^3-1)\) and a sector of aperture \(2\pi /3\).

Let \(c=\sqrt {2}-1\). Use the method of steepest descent to show that \[ \int _{-\infty }^{\infty } e^{ikx}(1+x^2)^{-k}\,dx \sim \sqrt {\frac {\pi (1-c)}{k}}\,e^{-ck}(2c)^{-k} \] as \(k\to \infty \). Hint: You do not need to work out the exact steepest descent contour, but give a sketch of what it looks like. You do not need to give full justification for the deformation from the original contour to the steepest descent contour, but at least write down an idea of why it should be true.

Complete the following:

1. (10 pts) Use de Moivre’s Theorem to show that the two expressions, \[ \cos (n\theta ),\qquad \frac {\sin ((n+1)\theta )}{\sin \theta }, \] for arbitrary \(n\in \mathbb {N}\), are both degree-\(n\) polynomials of \(\cos \theta \).
2. (10 pts) Prove the triangle inequality: Given \(z_1,z_2\in \mathbb {C}\), then \[ \bigl ||z_1|-|z_2|\bigr |\le |z_1+z_2|\le |z_1|+|z_2|. \]

1. (10 pts) Show that \[ \int _{-\infty }^{\infty }\frac {\cos x}{x^2-\frac {\pi ^2}{4}}\,dx=-2. \]
2. (10 pts) Compute the complete infinite asymptotic expansion of \[ I(k)=\int _0^\pi e^{-kt^2}t^{-3/4}\sin t\,dt. \]

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.

Compute the complete asymptotic expansion in powers of \(k\) of \[ I(k)=\int _0^\infty e^{-kt^2}\frac {\sin (t^4)}{t^{7/2}}\,dt,\qquad k\to \infty . \]

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.

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.

Use a sector contour with radius \(R\) centered at the origin with angle \(0\le \theta \le 2\pi /5\) to find, for \(a>0\), \[ \int _0^\infty \frac {dx}{x^5+a^5}=\frac {\pi }{5a^4\sin (\pi /5)}. \]

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

Evaluate \[ I=2\int _{-\infty }^{\infty }\frac {x\sin x}{x^2+4}\,dx. \] You must indicate the contour of integration and treat integration over each contour.

Evaluate \[ I=\int _{-\infty }^{\infty }\frac {\cos (2x)}{(x+1)^2+4}\,dx. \] You must clearly show the contour of integration and treat integration over each contour separately.

Consider the distribution \(u\in \mathcal {D}'(\mathbb {R})\) defined by \[ \langle u,\phi \rangle =\int _{-\infty }^{\infty }\lvert x\rvert \phi (x)\,dx, \qquad \phi \in \mathcal {D}(\mathbb {R}). \] Find its derivative, \(\partial u\).

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} \]

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.

Integrate \[ \begin {array}{rcl} \text {(a)}&\displaystyle \int _0^{2\pi }\frac {d\phi }{(p+q\cos \phi )^2} &\text {(parameters }p>q>0\text {)},\\[2.2ex] \text {(b)}&\displaystyle \int _{-\infty }^{\infty }\frac {e^{\alpha x}}{1+e^x}\,dx &\text {(parameter }\alpha \in (0,1)\text {)},\\[2.2ex] \text {(c)}&\displaystyle \int _0^{\infty }\sin (a^2x^2)\,dx &\text {(positive parameter }a\text {)}. \end {array} \]

Integrate

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

[Explain the logic (in particular, the choice of contour), but do not worry about getting the exact numbers; \(\alpha \) is a real parameter.]

Evaluate the integrals

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

Consider the distribution \(u\in \mathcal {D}'(\mathbb {R})\) defined for \(\phi \in \mathcal {D}(\mathbb {R})\) by \[ \langle u,\phi \rangle =\int _{-\infty }^{\infty }\lvert x\rvert \phi (x)\,dx. \] Find its derivative \(\partial u\).

Integrate \[ \int _{-\infty }^{\infty }\frac {\sin \alpha x}{\sinh \pi x}\,dx. \]

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

Evaluate

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

Evaluate the integrals:

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

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

Evaluate the integrals:

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

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 \[ I(s)=\int _0^1 \ln t\, e^{ist}\,dt, \qquad s\text { is real and }s\to +\infty . \]

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

Use Fourier transforms to 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 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).

The windowed Fourier transform of a function \(f(x)\in L^2(-\infty ,\infty )\) is defined by \[ Gf(\omega ,\mu )=\int _{-\infty }^{\infty }g(x-\mu )f(x)e^{-\omega x}\,d\mu \,dx. \] Suppose \(g(x)\) is a real valued function. State and verify the formula for the reconstruction of \(f(x)\) from \(Gf\), including any additional conditions on \(g(x)\). You may assume the validity of the Fourier transform.

1. Find the first term of the asymptotic representation of \[ n!=\int _0^\infty \exp (-t)t^{a-1}\,dt \] for large \(n\). (This approximation is called Stirling’s formula.)
2. What is the order of the error term and how can this be rigorously justified? (Invoke the appropriate theorem.)

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.

1. Show that a test function \(\psi (x)\) is of the form \(\psi =(x\phi )'\) where \(\phi \) is a test function if and only if \[ \int _{-\infty }^{\infty }\psi (x)\,dx=0 \quad \text {and}\quad \int _0^{\infty }\psi (x)\,dx=0. \]
2. Solve the following equation in the sense of distribution: \[ x^2\frac {d\phi }{dx}=0. \]

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

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 \(s\to +\infty \).

Consider the distribution \(g\) defined by \(\langle g,\phi \rangle =\int _{-\infty }^{\infty } e^{-|x|}\phi (x)\,dx\), for all test functions \(\phi \).

1. Calculate the second derivative \(g''\) explicitly. Prove \(g''\) is a distribution of order zero.
2. Find a differential equation for which \(g\) is a Green’s function.

Use contour integration to show that \[ \int _0^\infty \frac {x^\alpha }{(1+x)^2}\,dx=\frac {\pi \alpha }{\sin \pi \alpha } \] where \(0<\alpha <1\).

Discuss the flow pattern associated with the complex potential \[ \Omega (z)=Q_0z+\frac {\overline {Q_0}a^2}{z}+\frac {i\gamma }{2\pi }\log z \] with \(a>0\), \(\gamma \) real and \(Q_0=U_0+iV_0\). Sketch the flow when \(\gamma =0\). Given that the complex force on a cylindrical obstacle with boundary \(C\) is \[ \overline {F}=\frac {1}{2}i\rho \int _C\left (\frac {d\Omega }{dz}\right )^2\,dz, \] where \(\rho \) is the density of fluid, determine the lift when \(\gamma \ne 0\).

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.

Show that \[ \int _0^\infty \frac {\cosh ax}{\cosh \pi x}\,dx=\frac {1}{2}\sec \left (\frac {a}{2}\right ), \qquad |a|<\pi \] Use a rectangular contour with corners at \(\pm R\) and \(\pm R+i\).

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. Using Jordan’s lemma show that \[ \int _0^\infty \frac {\cos ax}{1+x^2}\,dx=\frac {\pi }{2}e^{-a}. \]
2. Show that \[ \int _0^{2\pi }\frac {d\theta }{A+B\sin \theta }= \frac {2\pi }{\sqrt {A^2-B^2}} \] where \(A^2>B^2\), \(A>0\).

Use contour integration to show that \[ \int _0^\infty \frac {x^\alpha }{(1+x)^2}\,dx=\frac {\pi \alpha }{\sin \pi \alpha } \] where \(|\operatorname {Re}\alpha |<1\).

Use Fourier transforms to show that the solution of the diffusion equation \[ \frac {\partial \phi }{\partial t}=\frac {\partial ^2\phi }{\partial x^2},\qquad -\infty <x<\infty \] with initial data \(\phi (x,0)=h(x)\) takes the form \[ \phi (x,t)=\int _{-\infty }^{\infty } g(x-x',t)h(x')\,dx' \] where \[ g(x,t)=\frac {e^{-x^2/4Dt}}{2\sqrt {\pi Dt}}. \] Determine \(\lim _{t\to 0}g(x,t)\).

1. Evaluate the following real integrals by contour integration: \[ \int _0^{2\pi }\frac {d\theta }{(5-3\sin \theta )^2}. \]
2. Let \[ e^{t/2(z-1/z)}=\sum _{n=-\infty }^{\infty }J_n(t)z^n. \] Show from the definition of Laurent series and properties of complex integration that the Bessel function \[ J_n(t)=\frac {1}{\pi }\int _0^\pi \cos (n\theta -t\sin \theta )\,d\theta . \]

Consider the wave equation \[ \frac {\partial ^2\phi }{\partial t^2}=\frac {\partial ^2\phi }{\partial x^2},\qquad 0\le x\le l \] with the following boundary conditions \[ \begin {gathered} \phi (x,0)=0,\qquad \frac {\partial \phi }{\partial t}(x,0)=0,\\ \phi (0,t)=0,\qquad \phi (l,t)=1. \end {gathered} \]

1. Show that the Laplace transform of the solution \(\widehat {\Phi }(x,s)\) is given by \[ \widehat {\Phi }(x,s)=\frac {\sinh sx}{s\sinh sl}. \]
2. Obtain the solution \(\phi (x,t)\) by inverting the Laplace transform using a Bromich integral and show that it can be expressed as the infinite series \[ \phi (x,t)=\frac {x}{l}+\sum _{n=1}^{\infty }\frac {2(-1)^n}{n\pi } \sin \left (\frac {n\pi x}{l}\right )\cos \left (\frac {n\pi t}{l}\right ). \]

1. Using Jordan’s lemma show that \[ \int _0^\infty \frac {\cos ax}{1+x^2}\,dx=\frac {\pi }{2}e^{-a}. \]
2. Show that \[ \int _0^{2\pi }\frac {d\theta }{A+B\sin \theta }= \frac {2\pi }{\sqrt {A^2-B^2}} \] where \(A^2>B^2\), \(A>0\).

Use contour integration to show that \[ \int _0^\infty \frac {x^\alpha }{(1+x)^2}\,dx=\frac {\pi \alpha }{\sin \pi \alpha } \] where \(|\operatorname {Re}\alpha |<1\).

Use Fourier transforms to show that the solution of the diffusion equation \[ \frac {\partial \phi }{\partial t}=\frac {\partial ^2\phi }{\partial x^2},\qquad -\infty <x<\infty \] with initial data \(\phi (x,0)=h(x)\) takes the form \[ \phi (x,t)=\int _{-\infty }^{\infty } g(x-x',t)h(x')\,dx' \] where \[ g(x,t)=\frac {e^{-x^2/4Dt}}{2\sqrt {\pi Dt}}. \] Determine \(\lim _{t\to 0}g(x,t)\).

Evaluate \[ \int _0^\infty \frac {\sqrt {x}}{x^2+1}\,dx. \]

1. Find the leading order behavior of \[ G(x)=\int _0^\infty \cos \left (x\left (\frac {t^3}{3}-t\right )\right )\,dt \] for large \(x\).
2. Outline the method to find more terms in the asymptotic expansion of \(G(x)\).

Useful information: \[ \int _{-\infty }^{\infty }\cos (ax^2)\,dx =\int _{-\infty }^{\infty }\sin (ax^2)\,dx =\sqrt {\frac {\pi }{2a}}. \]

Suppose the function \(\phi (x)\) is Lipschitz continuous on the unit interval \(0<x<2\pi \) and that \(\lim _{x\to 0}\phi (x)=\phi (0^+)\) exists. Suppose \(\int _0^\infty \frac {\sin x}{x}\,dx=\alpha \) is known. Evaluate \(\lim _{k\to \infty }\int _0^{2\pi }\phi (x)\frac {\sin kx}{x}\,dx\) (with \(k\) restricted to be integer-valued).

Evaluate \[ \int _{-\infty }^{\infty }\frac {\sin x}{x}\,dx. \]

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, \(-1<\alpha <1\).

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

Evaluate by contour integration \[ I=\int _0^\infty \frac {x\sin 3x}{4+x^2}\,dx. \]

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