Articles of complex analysis

Integration using residues

For the following problem from Brown and Churchill’s Complex Variables, 8ed., section 84 Show that $$ \int_0^\infty\frac{\cos(ax) – \cos(bx)}{x^2} \mathrm{d}x= \frac{\pi}{2}(b-a)$$ where $a$ and $b$ are positive, non-zero constants, by integrating about a suitable indented contour. The contour in question is the upper half of an annulus bisected by the $x$-axis with an outer radius […]

Fourier Transform on Infinite Strip Poisson Equation

Im trying to solve the following Poisson equation: $$u_{xx} + u_{yy} = \exp(-x^2)\ \text{for}\ x \in (-\infty, \infty)\ \text{and}\ y \in (0,1)$$ $$u(x,0) = 0,\ u(x,1) = 0$$ $$u(x,y) \to 0\ \text{uniformly as}\ |x| \to \infty\ \text{(i.e. compact support).}$$ I want to solve this using the Fourier Transform. I’ve tried taking the Fourier Transform with […]

Evaluating a trigonometric integral by means of contour $\int_0^{\pi} \frac{\cos(4\theta)}{1+\cos^2(\theta)} d\theta$

I am studying for a qualifying exam, and this contour integral is getting pretty messy: $\displaystyle I = \int_0^{\pi} \dfrac{\cos(4\theta)}{1+\cos^2(\theta)} d\theta $ I first notice that the integrand is an even function hence $\displaystyle I = \dfrac{1}{2} \int_{-\pi}^{\pi} \dfrac{\cos(4\theta)}{1+\cos^2(\theta)} d\theta $ Then make the substitutions $\cos(n\theta) = \dfrac{e^{in\theta}+e^{-in\theta}}{2}$, and $z=e^{i\theta}$ to obtain: $\displaystyle I = […]

Evaluate the Bessel Function $J = \int^{2\pi}_{0}{e^{\cos x}}{\cos(2x – \sin x)}\, dx$

I need to evaluate the following definite integral: $$J = \int^{2\pi}_{0}{e^{\cos x}}{\cos(2x – \sin x)}\, dx$$ I have attempted basic variable substitution and expanding the cosine term, but I have not been able to find an indefinite integral. I believe the best strategy would be to use contour integration, but I am not sure on […]

Intersections of the level curves of two (conjugate) harmonic functions

Let $f:\mathbb{R}^2\rightarrow\mathbb{R}$ with $f(x,y)=e^{-x}(x\sin y-y\cos y)$. 1 Let $g$ be one of the conjugate harmonics of $f$ on $\mathbb{R}^2$ and assume the level curves of $f$ and $g$ intersect.How do I show that the level curves intersect at right angles (by calculating)? 2 What is the conceptual explanation behind the right angle intersection? What I […]

How to do contour integral on a REAL function?

Suppose we are given the problem: Evaluate: $$\int_{0}^{\infty} \frac{1}{x^6 + 1} dx$$ Where $x$ is a real variable. A real variable function (no complex variables). I was reading Schaum’s outline to this solution and it defines: $f(z) = \displaystyle \frac{1}{z^6 +1}$ And it says consider a closed contour $C$ Consisting of the line from $-R$ […]

Proving that $ e^z = z+\lambda$ has exactly $m+n$ solutions $z$ such that $-2\pi m<\Im z<2\pi n$

I need to prove that for $\lambda\in \mathbb{C}$ and for $m,n\in\mathbb{Z}$ large enough, the equation: $$ e^z = z+\lambda$$ has exactly $m+n$ solutions $z$ such that $-2\pi m<\Im z<2\pi n$, where $\Im z$ denotes the imaginary part of $z$. I thought about looking at the rectangle $\pm R + 2\pi in, \pm R – 2\pi […]

What is the Hadamard's Factorization of a function that has a finite number of zeros

Hadamard’s Factorization Theorem. Let $f$ be an entire function of order $ρ=1$. Then $f(s)=P(s)exp(Q(s))$, where $P(s)$ is a canonical product with the same zeros as $f$ and $Q$ is a polynomial of degree less than or equal to $ρ=1$. In this case we have $|f(s)|≤_{ε}exp(|s|^{ρ+ε})$ for all $ε>0$. We have $f(s)=exp(Q(s))∏_{k=1}^{∞}(((s_{k}-s)/(s_{k})))exp( (s/(s_{k})))$ where $s_{k}$ are […]

Describing all holomorphic functions such that $f(n)=n$ for $n \in \mathbb{N}$

This question is inspired by a somewhat simpler one. The question is: how can we classify all holomorphic functions $f:\mathbb{C}\rightarrow\mathbb{C}$ satisfying the property $\forall n \in \mathbb{N} \quad f(n)=n $? If we have $g:\mathbb{C}\rightarrow\mathbb{C}$ such that $g\big|_\mathbb{N}\equiv 0$, then $f(z)=z+g(z)$ satisfies the criterion. Conversly, given such $f$ and defining $g(z)=f(z)-z$, we get $g\big|_\mathbb{N}\equiv 0$. So, […]

Prove that a complex valued polynomial over two variables has infinitely many zeroes

This is a homework question that I am struggling with. Given a polynomial over the complex numbers in two variables, show that the polynomial has infinitely many zeroes. So let’s say that the polynomial is a functions of $u$ and $v$. Let’s consider the polynomial as a function of $u$, with $v$ as a parameter: […]