# 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, ...

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### Long polynomial expansion with 34 roots

This is a very tricky problem, I just need a few hints. I think the $(-x^{17})$ is also there for a specific trick. In the end if it is $ax^{17}$, I see that $a = 17 - 1 + 1 = 17$. Also, another ...
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### What does $\Bbb R/2\pi$ for a set mean?

I simply cannot figure out what this means. I read this on an article about the scalar product of $2\pi$ periodic functions. it says that < f,g > goes from $\Bbb R/2\pi \to \Bbb C$ (complex) Do ...
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### A certain Complex line integral

In evaluating the line integral of $\frac{dz}{z-2}$ around the circle $|z-1|=5$ , and also around the square with vertices $3+3i ,3-3i,-3+3i,-3-3i$, I obtain zero in both cases however my textbook ...
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### Find the Laurent series about $z=0$

Let $f(z)=\cfrac{e^{-3z}}{z^2(z-2)^2}$, find the Laurent series about $z=0$. On the region $0<|z|<2$, I get $\cfrac{1}{(z-2)^2}=\displaystyle\sum_{n=1}^{\infty}\cfrac{nz^{n-1}}{2^{n+1}}$, ...
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### set of numbers satisfying a complex exponential equation

Here is the question: Using the principle branch definition of $z^i$ determine the set of all $z\in\mathbb{C}$ for which $(z^i)^2=(z^2)^i$. My ideas: I took the principle branch to be ...
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### Mapping the upper half plane conformally onto a semi-infinite strip,

Map the upper half y>0 of the z-plane conformally onto the semi-infinite strip u>0, $-\pi<v<\pi$ in the w-plane. I would like some hints for now, please. I'm not sure how to even get started ...
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### Is the limit $\lim\limits_{x\to\infty} {i}^{-x}$ equal to $0$, or doesn't exist?

Can someone show me if this limit exists: $$\lim_{x\to \infty} {i}^{-x}=0$$ or it doesn't exist? Here, $i$ is the unit imaginary part. Thank you for any help.
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### Best estimate using Cauchy integral formula: why is a circle the optimal path?

I once encountered this question from Ahlfors' Complex Analysis. An analytic function $f$ has the property that for $|z|<1$, $|f(z)|\leq \frac{1}{1-|z|}$. Find the best estimate of ...
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### How to remember/rederive the isomorphisms from the half planes to the unit disc

I know that $$z \mapsto \frac{z-i}{z+i}$$ maps the upper half plane to the unit disc, and $$z \mapsto \frac{z-1}{z+1}$$ maps the right half plane. Is there an intuitive way to construct such maps ...
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### When proving that f(z) is a polynomial, is it enough to consider just one point instead of keeping z arbitrary?

I think so - but I'd rather ask the MSE community too. Say I am given the bound |f(z)| < $|z|^3$, and that f is entire. Show f must be a polynomial. I used Cauchy's Integral Formula for ...
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### Slopes of curves from complex derivative [closed]

Show that the slopes of the level curves$$u(x,y)=\text{constant} \ \ \text{and} \ \ v(x,y)=\text{constant}$$ are respectively given by $$\cot(\arg(f'(z))) \ \ \text{and} \ \ -\tan(\arg(f'(z)))$$ If ...
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### What does “bounded away from zero” actually mean?

For example, is $f(z) = 1/z$, on the set $0<z<1$ "bounded away from zero"?
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### Proof or Counterexample:Is every open connected set $D \subset \mathbb C$ is a domain of holomorphy?

Def: An open set $D \subset \mathbb C^n$ is called a domain of Holomorphy if there exists a holomorphic function $f$ on $D$ such that $f$ cannot be extended to a bigger set. Is every non empty open ...
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### Compute definite integral

Question: Compute $$\int_0^1 \frac{\sqrt{x-x^2}}{x+2}dx.$$ Attempt: I've tried various substitutions with no success. Looked for a possible contour integration by converting this into a rational ...
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### Existence of unique circle passing through interior points of unit disk meeting the boundary orthogonally

I am a self-studies and this is a hw problem from a complex analysis scourse I've been doing. The problem set pertains to the topic Automorphism Groups and has a high concentration of fractional ...
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### How do I determine if a given function is entire? [closed]

Consider the three functions $\displaystyle e^{\frac{r}{\ln r}}$, $\displaystyle e^r$, and $\displaystyle e^{r\ln r}$, where $r = |z|$. Note that these are not constant functions. Can someone ...
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### Reducing multi-variable functions to a composition of 1- or 2-variable functions

There are some special functions of 3 or more complex variables that are analytic in some domain (a region in $\mathbb C^n$) with respect to each variable. To give some examples: the incomplete beta ...
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### Computing the residue of $\frac{z-2}{z^2} \sin\left(\frac{1}{1-z}\right)$ for $z = 1$.

Consider the function $$f(z) = \frac{z-2}{z^2} \sin\left(\frac{1}{1-z}\right)$$ We have that $0$ is a double pole and $1$ is a single pole (essential singularity) of $f$. It is simple to compute ...
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### For given $t$ and $x$ and $y$, is there at least one $f$ such that $\cos ft = x, \sin ft =y$?

Suppose that $t$, $x$ and $y$ are given and are all in $\mathbb{R}$. Is there always at least one $f$ such that $\cos ft = x, \sin ft =y$? Edit: OK I forgot to add that given $x$ and $y$ are such ...
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### How to get sine term in Analytical continuation of $\zeta(s)$

I am able to prove the symmetric functional equation that Riemann gives in his paper, using Poisson Summation and properties of $\theta(x)$. The functional equation is given like so, ...
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### Proofing Analytic continuation and stationary increments of an exponential Family

In U.Küchler "Exponential Families of Stochastic Processes" 1997 Theorem 4.2.1 we have the following setup. Let $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq0})$ be a filtered measurable space. Let ...
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### Nontrivial homomorphisms from G to T

Let $G$ be a compact metric abelian group. $T$ be the circle group. Let $\mathcal{A}$ be the set of all finite linear combinations of continuous homomorphisms from $G \to T$. I want to show ...
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### Why the complex number system is not an ordered field [duplicate]

In high school, we are taught that we do not have $2i < 3i$, i.e., the complex number system is not an ordered field. (Real number, for example, is an ordered field. For example, $2 < 3$). ...
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### A holomorphic function on a punctured disc has removable singularity iff it can be approximated by polynomials on a circle

Let $r>0$ and $f: D(0,2r)\setminus\{0\} \to \mathbb{C}$ be holomorphic, where $D(0,2r):= \{z \in \mathbb{C} \,:\, |z|<2r\}$. Show that f has removable singularity at $0$ iff ...
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### Computation of a certain integral

Assume that $\alpha>0, t \in R$. Compute the integral $\int_0^1(-1)^xx^{-\alpha-it}dx.$
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### Question on a parameter $\alpha_1$

Assume that $\alpha_1>0, \alpha_2>0, t_1 \in R, t_2 \in R$. Does the validity of the equality $\int_0^1(-1)^xx^{-\alpha_1-it_1}dx/\int_0^1(-1)^xx^{-\alpha_2-it_2}dx=0$ implies that ...
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Let $p$ be a polynomial in 1-complex variable.Suppose all zeroes of $p$ are in the upper half plane $H=\{z \in \mathbb{C} :Im(z)> 0 \}$. Then which of the following are true? $Im \frac {p'(z)}{ ... 1answer 37 views ### Integral formula involving logarithms and the zeros of a holomorphic function I have the following formula I´d like to prove: Given a holomorphic function$f:U\to \mathbb C$such that$\overline{D_r(0)}\subset U$,$f(0)\neq 0$and$f(z)\neq 0$for$z\in \partial D_r(0)$, we ... 2answers 96 views ### Cauchy Riemann equations necessary and sufficient condition? I was taught that$f(z)$is differentiable at$z_0=x_0+y_0$iff Cauchy Riemann equations hold at$(x_0,y_0)$. However, I was shown this example:$f(z)=\frac{\operatorname{Re}(z) \cdot ...
$f(z)=\frac{e^z}{z+1}$ I know that $0$ and $\infty$ are two asymptotic values of the above function. Question:Does there exist another asymptotic values other than $0$ and $\infty$ ?
### Where is $f(z)=Im(z^2)$ differntiable?
Where is $f(z)=Im(z^2)$ differntiable? So what I tried $f(z)=f(x+iy)=Im((x+iy)^2)=2x^2y^2$ Then by Cauchy Riemann: $U_x=V_y \rightarrow 8y^2x=0$ and $U_y=-V_x \rightarrow 8x^2y=0$. Then, my ...