This is for an assignment, describing the procedure is most beneficial for me, rather that solely computing the result. I want to evaluate the following integral: $$\int_C\text{Re}\;z\,dz\,\text{ from }-4\text{ to } 4$$ Along the line segments from $-4$ to $-4-4i$ to $4-4i$ to $4$, e.g: So I want to evaluate three integrals of the above […]

I need to determine where the following function is differentiable and holomorphic in $\mathbb C$: $$f(z)=(z-3)^i$$ I have the derivative as $df/dz= i(z-3)^{-1+i}$. The answer in my book says f is differentiable and holomorphic on $\mathbb C$ where $y\neq0$ and $x>3$. I don’t see where this comes from. wolframalpha plotted the derivative and I can […]

Let $S$ denote the Riemann Sphere. Recall that a Mobius transformation is a function $f:S \to S$ defines as $z \to \frac {az+b}{cz+d}$ where $a,b,c,d \in \mathbb C$ with $ad-bc=1$. What is the motivation to study Mobius Transformation? Why should one look at the map defined in the above way?

Can someone give me an example of where the Lagrange inversion theorem is applied in such a way it inverts a formal series? For example, say I have $$\sum_{i>-1} a_it^i = u.$$ Can someone show me the step by step process by which $$\sum_{i>-1}b_iu^i = t$$ is obtained. I can seem to find any links […]

Consider the following boundary value problem on the infinite strip $(-\infty,\infty)\times[0,1]$ w/periodic conductivity $\gamma(x,y)=\gamma(x+2\pi,y)>0$: $$\begin{cases} \operatorname{div}(\gamma\nabla u)=(\gamma u_x)_x+(\gamma u_y)_y=0,\\ u(x+2\pi,y)=\mu u(x,y),\\ u(x,1)=0,\\ u(x,0)=0 \mbox{ or } u_y(x,0)=0. \end{cases}$$ Let’s denote the set of possible values of the eigenvalue parameter $\mu$, for which the system has a solution $\mu_\gamma$. Does there exist a conductivity $\alpha(y)$ on […]

In my complex analysis book, there is an example where I am asked to compute $\int_\Gamma1/z \, dz$ for two cases: in both of them, $\Gamma$ is a curve going from $-i$ to $i$ in the complex plane. However, in the first case, $\Gamma$ lies in both the first and fourth quadrants, crossing the positive […]

Setting: Let $(x_n)$ be Cauchy in $\ell^2$ over $\mathbb{F} = \mathbb{C}$ or $\mathbb{R}$. I’m trying to show that $(x_n) \rightarrow x \in \ell^2$. That is, I’m trying to show that $\ell^2$ is complete in a particular way outlined below. I only used the first few steps of the proof because once I understand the third […]

I need some help with the following problem from Ahlfors’ Complex Analysis. Problem: Find a single Möbius transformation $\phi$ (that is, a map of the form $\phi(z) = \dfrac{az + b}{cz + d}$, where $a,b,c,d$ are complex numbers) that maps the circles $|z| = 1$ and $\left|z – \frac{1}{4}\right| = \frac{1}{4}$ to concentric circles. Infinite […]

Compute $\displaystyle\int_0^\infty \frac{dx}{1+x^3}$ by integrating $\dfrac{1}{1+z^3}$ over the contour $\gamma$ (defined below) and letting $R\rightarrow \infty$. The contour is $\gamma=\gamma_1+\gamma_2+\gamma_3$ where $\gamma_1(t)=t$ for $0\leq t \leq R$, $\gamma_2(t)=Re^{i\frac{2\pi}{3}t}$ for $0\leq t \leq 1$, and $\gamma_3(t)=(1-t)Re^{i\frac{2\pi}{3}}$ for $0\leq t \leq 1$. So, the contour is a wedge, and by letting $R\rightarrow \infty$ we’re integrating over one […]

Let $f$ be analytic in open unit disk, we need to show there exist $\{z_n\}$ with $|z_n|<1$ and $|z_n|\rightarrow 1$ then $f(z_n)$ is bounded. could any one give me Hints for this one?

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