Articles of complex analysis

Why is the MacLaurin series proof for eulers formula $ e^{i\theta} = \cos(\theta) + i\sin(\theta) $ valid?

The proof for this $$ e^{i\theta} = \cos(\theta) + i\sin(\theta) $$ using the MacLaurin series is all right for a high school level, but I dont understand why the series that has been derived for the reals should hold for complex numbers too. Could someone give a sufficient reason why it is correct to use […]

How to show set of all bounded, analytic function forms a Banach space?

I am trying to prove that set of bounded, analytic functions $A(\mho)$, $u:\mho\to\mathbb{C}$ forms a Banach space. It seems quite clear using Morera’s theorem that if we have a cauchy sequence of holomorph functions converge uniformly to holomorph function. Now i am a bit confused what norm would be suitable in order to make it […]

Proving that $\phi_a(z) = (z-a)/(1-\overline{a}z)$ maps $B(0,1)$ onto itself.

I want to prove that if $\phi_a: B(0,1) \to \Bbb C$ is given by $\phi_a(z) = (z-a)/(1-\overline{a}z)$ with $|a| < 1$, then $|\phi_a(z)| < 1$. Resist the itch on your finger urging you to close the question: I already took a look at this question and this one. I’m supposed to prove things in the […]

identity of $(I-z^nT^n)^{-1} =\frac{1}{n}$

I am trying to understand the identity $$(I-z^nT^n)^{-1} =\frac{1}{n}[(I-zT)^{-1}+(I-wzT)^{-1}+…+(I-w^{n-1}zT)^{-1}] \quad (*),$$ where $T \in \mathbb{C}^{n\times n},z\in \mathbb{C}$ and the spectral radius of $T$, $\rho(T)=\max\{|\lambda|: \exists v, Tv=\lambda v\}\leq 1$ and $|z|<1$ and $w$ is a primitive $n$th root of 1,i.e. $w =e^{i2\pi/n}$. I have tried using the identity $$[I-A]^{-1} = I+A+A^2+A^3+… $$ which holds when […]

how do you convert y” equation into Sturm-Liouville form

For $y = y(x)$, convert the following equation $$ y”- 2xy’ + 2vy = 0;$$ where $v$ is a constant, into a Sturm-Liouville form $$ Ly = r(x)(\lambda)y,$$ $\lambda $ is a number, where $$ L := \frac{d}{dx} [ p(x) \frac{d}{dx} ] + q(x).$$ Here $r(x) > 0$ is a weight. The trick is to […]

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 […]