Complex Integration and Real Integral Evaluation

Using residue calculus evaluate

Compute the following integral using methods of complex analysis.

Prove the following version of Morera’s theorem. If \(f:\mathbb {D}\to \mathbb {C}\) is a continuous function such that for every rectangle \(R\subset \mathbb {D}\)

then \(f\) is holomorphic. Note: It does not follow from the assumption in any obvious way that integrals over triangles are 0.

Let \(\Omega \) be a bounded domain in \(\mathbb {C}\) and \(f_n:\Omega \to \mathbb {C}\) a sequence of holomorphic functions such that \[ \int _\Omega |f_n-f_m|^2\,dA(z)\to 0\text { as }n,m\to \infty . \] Here \(dA(z)\) is the Lebesgue area integral. Show that there is \(f\) holomorphic on \(\Omega \) such that \(f_n\to f\) locally uniformly on \(\Omega \).

Let \(a, b > 0\). Using the residue calculus evaluate the improper integral

Using the residue calculus evaluate the integral

Let \(U\) be a domain in \(\mathbb {C}\). Consider the following collection of functions \[ X=\{f:U\to \mathbb {C}: f\text { holomorphic and }\int _U |f|^2\le M\} \]

where \(\int _U\) is the standard Lebesgue area integral. Show that \(X\) is compact with respect to the topology of uniform convergence on compact subsets of \(U\) (Note: this topology is metrizable so you can show sequential compact- ness).

Compute the following integral using the techniques of complex analysis \[ \int _0^\infty \frac {\log (x)}{2+x^2}\,dx. \]

Use the methods of complex analysis to compute \[ \int _{-\infty }^{\infty }\frac {dx}{(1+x^2)^3}. \]

Evaluate the integral by the method of residues: \[ \int _0^\infty \frac {x^{1/3}}{1+x^2}\,dx. \]

Let \(f:[0,\infty )\to \mathbb R\) be defined by \(f(x)=e^{-x}\). Explain why \(f\) is Lebesgue integrable and compute its integral.

Using the methods of complex analysis compute \[ \int _0^\infty \frac {1}{x^4+16}\,dx. \]

Using the methods of complex analysis, compute \[ \int _0^\infty \frac {x^2}{x^4+1}\,dx \] and simplify your answer.

Let \(f:[0,\infty )\to \mathbb R\) be defined by \(f(x)=e^{-x}\). Explain why \(f\) is Lebesgue integrable and compute its integral.

Use the methods of complex analysis to compute \[ \int _{-\infty }^{\infty }\frac {2}{(x^2+4)(x^2+1)}\,dx. \]

Let \(f:[0,\infty )\to \mathbb R\) be defined by \(f(x)=e^{-x}\). Explain why \(f\) is Lebesgue integrable and compute its integral.

Using methods of complex analysis, evaluate \[ \int _0^\infty \frac {x^2}{x^4+x^2+1}\,dx. \]

Evaluate the integral by the method of residues: \[ \int _0^\infty \frac {x^2}{x^4+5x^2+6}\,dx. \]

Evaluate the integral by the method of residues: \[ \int _0^\infty \frac {x\sin x}{x^2+a^2}\,dx, \] for \(a\in \mathbb R\).

Using the residue theorem, find the integral \[ \int _{-\infty }^{\infty }\frac {\sin x}{x^2+4x+20}\,dx. \]

What is \[ \int _{-\infty }^{\infty }\frac {(\cos x)^2}{x^2+5}\,dx? \]

Evaluate \[ \int _C \frac {1}{(z+i)(z-1)}\,dz \] where \(C\) is pictured on the next page:

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Evaluate the integral \[ \int _0^\infty \frac {x^2+1}{x^4+1}\,dx. \]

Evaluate the integral \[ \int _{-\infty }^{\infty }\frac {x\sin x}{x^2+2x+10}\,dx \] using the residue theorem.

Evaluate the integral \[ \int _{-\infty }^{\infty }\frac {x\cos x}{x^2-2x+10}\,dx \] using the residue theorem.

Let \(n\) be a positive integer. Use a contour integral to compute \[ \int _0^{2\pi }\cos ^{2n}(\theta )\,d\theta \] in terms of \(n\).

Let \(f\in L^1(\mathbb R)\) such that \[ \int _{-\infty }^{\infty }|x\hat f(x)|\,dx<\infty \] Prove that there is a differentiable function \(g\) on \(\mathbb R\) such that \(f=g\) almost everywhere.

Let \(f:\Omega \to \mathbb C\) be a complex-valued function on an open domain \(\Omega \subset \mathbb C\). Assume that for all closed contours \(\gamma \) that consists of line segments parallel to the real or imaginary axes \[ \int _\gamma f(z)\,dz=0. \] Show that there exists a holomorphic function \(F:\Omega \to \mathbb C\) such that \(F'(z)=f(z)\).

Compute the integral \[ \int _0^\infty \left (\frac {\sin x}{x}\right )^2\,dx. \]

Calculate the integral \[ \int _{-\infty }^{\infty }\frac {\cos x}{x^2+4}\,dx. \]

Provide a rigorous computation of the sum of the following series: \[ \sum _{n=1}^{\infty }\frac {1}{n^4}. \] (Hint: one possible approach is to consider the integral of \(f(z)=\frac {\pi }{z^4}\cot (\pi z)\) over an appropriate sequence of contours.)

Determine the region of values of \(z\) for which \[ f(z)=\int _0^\infty \frac {e^{tz}}{1+t^2}\,dt \] defines a holomorphic function. (Hint: one approach is to use the holomorphicity criterion of Morera’s Theorem. If you pursue this approach, make certain to justify any application of Fubini’s theorem.)

Let \((X,\mathfrak M,\mu )\) be a positive measure space, and let \(f\ge 0\) be a measurable function on \(X\). Define \(g(t)=\mu (\{x:f(x)\ge t\})\). Show that \[ \int _X f(x)\,d\mu (x)=\int _0^\infty g(t)\,dt. \]

Evaluate the integral \[ \int _0^\infty \frac {\sin ax}{x(x^2+b^2)}\,dx \] where \(a\) and \(b\) are real numbers and \(b\ne 0\).

Evaluate the integral: \[ \int _0^\infty \frac {dx}{1+x^3} \]

Let \(\mu \) be a measure on a measure space \(X\) and let \(f\ge 0\) be a measurable function on \(X\). Prove that \[ \int _X f(x)\,d\mu (x)=\int _0^\infty \phi (y)\,dy \] where \(\phi (y)=\mu (\{x:f(x)\ge y\})\).

Compute \[ \int _0^\infty \frac {x^2}{x^4+1}\,dx. \]

Using methods of complex analysis, evaluate \[ \int _0^\infty \frac {x^2\,dx}{x^4+x^2+1} \]

Let \(f\) be holomorphic in an open set containing the disk \(D=\{z\in \mathbb C:|z|\le r\}\) and suppose that the only zero of \(f\) in \(D\) is a simple zero at \(a\) belonging to the interior of \(D\). If \(C_r\) is the positively oriented contour \(|z|=r\) show that \[ \int _{C_r}\frac {1}{f(z)}\,dz=\frac {2\pi i}{f'(a)}. \]

Use methods of complex analysis to find \[ \int _{-\infty }^{\infty }\frac {dx}{x^4+1}. \]