# Tagged Questions

The theory of functions of one complex variable with an emphasis on the theory of complex analytic (or holomorphic) functions of one complex variable. Typical topics include: Cauchy's integral formula, singularities, poles, meromorphic functions, Laurent and Taylor series, maximum modulus principle, ...

15 views

### Contour Integration problem solve

Let C be the circle $z=2+e^{i\theta}$, where $0\le\theta\le2\pi$, Evaluate $$\int_{c}\frac{\sin z}{z^2+2z} \ dz.$$ Thank you for cooperation.
12 views

### If $x(z)$ and $y(z)$ are analytic with $x(0) = 0 = y(0)$ then $x(z)^{y(z)} \to 0$ as $z \to 0$

I'm a programmer and my math is a little rusty, but usually sufficient for my needs. However, I came across the following statement in the exp(3) manual page of ...
36 views

### Is this simple looking complex expression valid always?

$$z^{a+ib} = z^a*z^{ib} \hspace{2mm} \forall z\in \mathbb{C}$$ In high school I was always taught to see the + in complex numbers as analogous to that is reals. However can it be proven to be ...
26 views

### Contour integrals on complex analysis

Let C be that part of the circle $z=e^{i\theta}$, where $0\le\theta\le\frac\pi2$. Evaluate $\int_{c}\frac{z}{i}dz$. This is my first time posting my question here. I'm really poor in writing English. ...
14 views

36 views

### Show limit does not exist

Show that this limit does not exist, z is Complex number: $\lim_{z \to -1 } \frac{1}{z^3}\sin (\frac{z}{z+1})$
34 views

35 views

45 views

### Let $f:\mathbb{C} \to \mathbb{C}$ be an non constant entire function such that $f(1-z)+f(z)=1$ for all $z\in \mathbb{C}$.

Let $f:\mathbb{C} \to \mathbb{C}$ be a non constant entire function such that $f(1-z)+f(z)=1$ for all $z\in \mathbb{C}$. Then prove that $f$ is surjective. It can be solved trivially by Picard's ...
34 views

70 views
+100

### $\log(e^z - i)$ as a holomorphic function in $\mathbb{D}$

I'm learning complex analysis, specifically holomorphic functions, and need help with the following exercise: Examine if the function $\log(e^z - i)$ can be defined as a holomorphic function in ...
38 views

### Conformal holomorphic mapping from disc to square

Let $f$ be a holomorphic map from the unit disc $\mathbb{D}$ to an open square $\mathbb{S}$ with its center at the origin. Given $$f(0) = 0, \qquad f'(z) \neq 0 \quad (z \in \mathbb{D})$$ prove that ...
25 views

Suppose that a function f is analytic in some domain $D$ which contains a segment of the x-axis and whose lower half is the reflection of the upper half with respect to that axis then $$\overline{f(z)... 2answers 41 views ### Argument of complex numbers If z=re^{i\theta} and w=\rho e^{i \phi}  are two complex numbers, then  arg(zw)=arg (z)+arg (w) But if z=-1 and w=-1, we get  0= 2\pi  which is not correct. So why it gives us this ... 1answer 33 views ### Proof of non-constant analytic functions Use the following theorem: "A function that is analytic in a domain D is uniquely determined over D by its values in a domain, or along a line segment, contained in D" to show that if f(z) ... 1answer 36 views ### Lebesgue integral of \frac{1}{\|\boldsymbol{x}-\boldsymbol{r}\|^2} on an infinite cylinder Let V\subset \mathbb{R}^3 be a solid infinite cylinder, or cylindrical shell, and let \boldsymbol{r}\in\mathbb{R}^3 be any point of the space. I intuitively suppose that the Lebesgue integral$$\...
Let $M=\left\{\left.\displaystyle z\mapsto\frac{az+b}{cz+d}\ \right|\ \ ad-bc\not =0\right\}$,$$p:GL(2,\mathbb C)\to M, \begin{bmatrix}a & b \\ c & d \end{bmatrix}\mapsto\frac{az+b}{cz+d}.$$ ...