Articles of complex analysis

What's a local angle?

When I was trying to understand the definition of conformal map I got confused. A conformal map is a function $f: U \to \mathbb C$ where $U \subset \mathbb C$ such that $f$ preserves local angles. But what is the definition of a local angle? I had no trouble imagining the meaning of oriented angle. […]

Heine borel theorem on the complex plane

I’m trying to understand this proof of the Heine-Borel theorem on the complex plane. I’m reading Lang’s Complex Analysis (page 22): I didn’t understand the converse. Why there is a convergent subsequence $\{z_{n_1}\}$? Why there is a convergent sub-subsequence $\{z_{n_2}\}$? Why $a+ ib\in S$? Thanks

Proving that $f(z) = \frac{1}{2} \left(z + \frac{1}{z}\right)$ is biholomorph on a certain set

Let $f: \mathbb{C} \backslash \{0\} \to \mathbb{C}$ be given by $$f(z) := \frac{1}{2} \left(z + \frac{1}{z}\right)$$ I first want to find the image of the set $H = \{z \in \mathbb{C}: |z| < 1, Im(z) > 0\}$ (the open upper half of the unit circle) regarding $f$. Next, I want to show that $f$ sends […]

If $f'/f=g'/g$ at every $1/n$ then $f=kg$ for some complex number $k$

Suppose $f(z)$ and $g(z)$ are analytic in domain $D$, $f(z)$ and $g(z)$ never vanish at any $z\in D$ and that $$ \frac{f'(z_n)}{f(z_n)}=\frac{g'(z_n)}{g(z_n)} $$ at a sequence of points $\{z_n\}$ converging to $z_0\in D$. Show that $f=Kg$ for some $K\in \mathbb{C}$. My Solution: As $f,g$ are analytic in $D$ and none of them have any zero […]

Trigonometric contour integral

I cannot figure out what I’m doing wrong: $$\int_0^{2\pi} \frac{1}{a+b\sin\theta} d\theta\quad a>b>0$$ $$\int_{|z|=1} \frac{1}{a+\frac{b}{2i}(z-z^{-1})} \frac{dz}{iz}$$ $$\int_{|z|=1} \frac{2i}{2ia+b(z-z^{-1})} \frac{dz}{iz}$$ $$\int_{|z|=1} \frac{2}{2iza+bz^2-b} dz$$ $$2iz_0a+bz_0^2-b=0$$ $$z_0=\frac{-2ia\pm\sqrt{-4^2+4b^2}}{2b}$$ $$z_0=\frac{a\pm\sqrt{a^2 – b^2}}{bi}$$ where only $z_0=\frac{a-\sqrt{a^2 – b^2}}{bi}$ is within $C$. So $Res(z_0) = \frac{-2b}{-2\sqrt{a^2-b^2}}$ So the integral is $\frac{2\pi bi}{\sqrt{a^2-b^2}}$ But this is wrong. The $bi$ should not be there. However […]

Biholomorphic Equivalence in $\mathbb{C}^n$

I am stuck at the following problem : For $n \geq 2$, are the sets $\Delta_n$ and $H_n$ biholomorphically equivalent ? Here, $ \Delta_n = \{ ( z_1, z_2, \ldots z_n) : |z_i| < 1, i = 1, 2, \ldots, n\} $ and $ H_n := \{ (z_1, z_2, \ldots, z_n) : \Im (z_1) > […]

Residue of sin(1/z) and its poles

I’m having difficulty determining the residue of sin(1/z). Not even WolframAlpha computes this. Is there any magic trick?

How does $\int_1 ^x \cos(2\pi/t) dt$ have complex values for real values of $x$?

This question is closely related to one I just asked here. I believe that it is just different enough to warrant another question; please let me know if it does not. In the question mentioned above, I was informed by Joriki that $$\int \cos\left(\frac{1}{x}\right) \mathrm{d}x = x \cos\left(\frac{1}{x}\right) + \operatorname{Si}\left(\frac{1}{x}\right)$$ where $$\mbox{Si}(u) = \int \frac{\sin(u)}{u} […]

Crazy calculation for winding numbers

Find the winding number around $z=-i, z=-1, z=0$ in the following figure. The purpose of this exercise is to complete a complex integral with singularities at the stated points. My attempt is that the winding number around $z=0$ is $1$, and that the winding numbers around $z=-i$ is zero, and $z=-1$ is $-1$. The reason […]

Real and Imaginary Parts of tan(z)

This is where I’m at: I know $$ \cos(z) = \frac{e^{iz} + e^{-iz}}{2} , \hspace{2mm} \sin(z) = \frac{e^{iz} – e^{-iz}}{2i}, $$ where $$ \tan(z) = \frac{\sin(z)}{\cos(z)}. $$ Applying the above, with a little manipulation, gives me: $$ \tan(z) = \frac{i\left(e^{-iz} – e^{iz}\right)}{e^{iz} + e^{-iz}}.$$ My thoughts are that I could use $e^{z} = e^{x+iy} = […]