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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74 views

Help with the integral $\int_{0}^{\infty}\frac{y^{2}e^{y}}{e^{sy}+e^{-sy}-2}dy$

I want to do the integral : $$I(s)=\int_{0}^{\infty}\frac{y^{2}e^{y}}{e^{sy}+e^{-sy}-2} \, \mathrm{d}y$$ $s$ being a complex parameter. I tried expanding the dominator of the integrand, but this way ...
0
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1answer
35 views

Residue of $\frac{e^{iz}}{z^2+4z+5}$ [closed]

I need to find the residue of $\dfrac{e^{iz}}{z^2+4z+5}$ at its singular points. How do I do that?
1
vote
1answer
485 views

Change of variable (translation) in complex integral

If I have a real integral, e.g. $\int f(x+2) \ dx$, I can substitute $y = x+2$, so $dy = dx$. But if my function is complex, am I still allowed to do this? In which cases I cannot apply a ...
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0answers
36 views

Show the integration with a complex variable

I want to show that there exists inverse Laplace transform, $f(t)$ of the function $F(\lambda)$. In other word, given $F(\lambda)$, existence of function $f(t)$ such that $$ ...
0
votes
1answer
40 views

Square root of a function

Let $D$ be a circular annulus in $\Bbb C$ with center at $0$. Put $v(z)=z$, for every $z\in D$. Show that $v$ has no square root measurable function. I think if define function $h$ such that ...
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0answers
29 views

A holomorphic line integral?

Let $f$ be a holomorphic function on a connected open set $U$ which contains the unit disc $D$, and suppose $|f(z)| \geq 1$ for $|z| = 1$. For $w \in D$, the argument principle tells us that ...
2
votes
3answers
75 views

find all $v(x,y)$ so that $f(x+iy)=u(x,y)+iv(x,y)$ is entire

I'm practicing to write down solutions clearly and thoroughly. Is this a proper answer to this exercise? How can it be improved? Let $u(x,y)=x^3-3xy^2$, find all $v(x,y)$ so that ...
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2answers
507 views

Simpler way to evaluate the Fourier transform of $\exp\left(i e^x\right)$?

I have the task to evaluate $|a(k)|^2$ with $$ a(k) = \int_{-\infty}^\infty \!dx\,\exp\left(i k x + i e^{x}\right).\tag{1}$$ The integral in (1) can be evaluated explicitly via the substitution ...
0
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0answers
32 views

When is a line integral of a holomorphic function holomorphic?

Let $U$ be a connected open set, $z_0 \in U$, and $g$ holomorphic on $U$. I know that if $U$ is simply connected, then $$G(z) := \int_{z_0}^z g(w)dw$$ is holomorphic on $U$, and $G'(z) = g(z)$. Here ...
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0answers
37 views

On the definition of modular forms

In many books, I see people defining modular forms to be holomorphic/meromorphic functions in the upper half plane such that it is invariant under the $|_k$ action of the group ...
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0answers
39 views

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 ...
2
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0answers
57 views

Find $\int_0^{2\pi} \frac{d\theta}{2\pi\cos^{2n}(\theta)} \ n=1,2,3,\dots$ via Residue Theorem

So the question is as follows: Use the Residue Theorem to calculate $$\int_0^{2\pi} \frac{1}{2\pi\cos^{2n}(\theta)} d\theta \quad\quad n=1,2,3,\dots.$$ Now I believe the first step would be to use the ...
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vote
0answers
28 views

If $f=u+iv:D\to \Bbb C$ is analytic on a domain D, is then the curves $u(x,y)=c_1$ and $v(x,y)=c_2$ intersect orthogonally?

If $f=u+iv:D\to \Bbb C$ is analytic on a domain D (an open connected subset of $\Bbb C$), is then the curves $u(x,y)=c_1$ and $v(x,y)=c_2$ intersect orthogonally, for any constants $c_1$ and $c_2$? ...
6
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1answer
66 views

Does there exists an entire function $f: \mathbb C \to \mathbb C$ which is bounded on real line and imaginary line?

Does there exists a nonconstant entire function $f: \mathbb C \to \mathbb C$ which is bounded on real line and imaginary line? Clearly,$ f(z)=sin(z)$ is an example of an entire function which is ...
3
votes
1answer
43 views

Why this map is a mobius transformation

Question: Let $D_2=\bar D(2,1)$ and $D_{-2}=\bar D(-2,1)$ be the closed disks of radius $1$ centered at $z=2$ and $z=-2$ in the complex plane, respectively. Set $X= \mathbb C-\{D_2 \cup D_{-2} \}$, ...
1
vote
1answer
21 views

Extending functions on proper subsets of $\mathbb C$ to functions on proper subsets of $S^2$.

There are a number of nice results about extending holomorphic and meromorphic functions from the complex plane $\mathbb C$ to the Riemann sphere $S^2$. See for instance Does entire function extend ...
3
votes
2answers
40 views

plot graph of function $f(z)=\frac{1+z}{1-z}$

I am not able to plot graph of function $f(z)=\frac{1+z}{1-z}$. can anyone tell me how to do this without using any software?
3
votes
2answers
68 views

Proving that the cross ratio is a Möbius transformation

I'm trying to show that given three distinct points $z_1,z_2,z_3\in\mathbb C$, the rational function $$ f(z) = \frac{(z-z_1)(z_2 - z_3)}{(z - z_3)(z_2 - z_1)} = \frac{(z_2 - z_3)z + (z_1z_3 - ...
4
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2answers
49 views

Complex integral with exponential and tangent

Suppose that $k \in \mathbb{R}.$ Evaluate as a function of $k$ the integral $$I(k) : = \int_{-\pi/2}^{\pi/2} e^{i \ k \ \mathrm{tan}(\phi)} d \phi.$$ Any suggestions on how to approach this problem? ...
6
votes
6answers
272 views

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 ...
4
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0answers
62 views

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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0answers
26 views

Is the Riemann–Liouville fractional derivative holomorphic in order?

If my understanding of complex analysis is correct then the arbitrary order generalization of Cauchy's formula for repeated integration $$(J^\alpha f) ( x ) = { 1 \over \Gamma ( \alpha ) } \int_0^x ...
3
votes
1answer
445 views

Derivation of poisson kernel for disk of radius $R$ from unit disk

Is there a way to derive poisson kernel for disk of radius $R$ from unit disk?
2
votes
2answers
66 views

Gaussian integral with a shift in the complex plane

$$ \int_{-\infty}^\infty e^{-(x+ia)^2} \text{d}x $$ where $a\in \mathbb{R}$. I don't know where to start but have reasons to believe the answer is $\sqrt{\pi}$. Namely $\int_{-\infty}^\infty ...
3
votes
1answer
54 views

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 $\cfrac{1}{(z-2)^2}=\displaystyle\sum_{n=1}^{\infty}\cfrac{nz^{n-1}}{2^{n+1}}$, ...
1
vote
1answer
44 views

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 ...
2
votes
1answer
38 views

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 ...
2
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1answer
63 views

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 ...
6
votes
1answer
65 views

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 ...
4
votes
2answers
62 views

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 ...
2
votes
1answer
67 views

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 ...
2
votes
2answers
96 views

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.
5
votes
1answer
640 views

Order of growth of the entire function $\sin(\sqrt{z})/\sqrt{z}$

Show that $$f(z)=\frac{\sin\sqrt z}{\sqrt z}$$ is an entire function of finite order $\rho$ and determine $\rho$. I observed that the two determinations of the square root differ only for the signum. ...
10
votes
1answer
87 views

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 ...
2
votes
1answer
56 views

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 ...
0
votes
2answers
37 views

Computing the residue of $\phi/\psi$ given conditions.

Let $\phi$ and $\psi$ be holomorphic functions around $z = a$, where $\phi(a) \neq 0$ and $a$ is a double root of $\psi(z) = 0$. Prove that the residue of $\phi(z)/\psi(z)$ at $z = a$ is: ...
1
vote
1answer
48 views

Borel Measures: Lusin

I'm trying to self-learn. Given the complex plane $\mathbb{C}$. Consider a Borel measure: $$\mu:\mathcal{B}(\mathbb{C})\to\mathbb{C}:\quad\mu\geq0$$ Regard a measurable: ...
4
votes
2answers
122 views

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"?
2
votes
2answers
67 views

How do I determine if a given function is entire?

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 ...
2
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0answers
26 views

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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0answers
17 views

Classification of isolated singularity by limit

Let $z_0$ be an isolated singularity of $f$ so there exist a punctured ball $B'$ centered in the singularity where $f$ is holomorphic. Let $f(z)=\sum_n a_n (z-z_0)^n$ be the Laurent series of $f$ in ...
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0answers
29 views

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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2answers
36 views

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 ...
2
votes
1answer
52 views

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, ...
3
votes
2answers
86 views

Definite integral (in the complex plane?)

I want to prove that $$\int_{0}^{\infty} \frac{dx}{1+x^b} = \frac{\pi}{b \sin(\pi/b)} \ ,$$ where $b\in (1,\infty)$. I thought about doing it in the complex plane since the integrand is a ...
4
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1answer
38 views

$f$ continuous on $\overline{D} \setminus \{1\}$, holomorphic and bounded on $D$. Then $f$ attains its supremum on the boundary

Let $D$ be the unit disc, $f$ continuous on $\overline{D} \setminus \{1\}$, holomorphic and bounded on $D$. The problem is to show that for all $z \in D$, $$|f(z)| \leq \sup\limits_{|\zeta| = 1, ...
3
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0answers
38 views

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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1answer
35 views

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 ...
3
votes
0answers
43 views

Is this function, a sum of one term and a convergent series, analytic?

$$(\frac{1}{z} + \sum z^n)$$ for 0<|z|<1. This is for complex variables. So, the series, convergent for the above domain of definition, always represents an analytic function. What about the ...
4
votes
1answer
79 views

Show: An entire function $g$ with $\vert g(x) \vert \to \infty$ for $|x| \to \infty$ is a polynomial.

This is part of an exercise sheet in complex analysis. It should by solvable by rather elementary methods like the main theorems of complex analysis. I succeded to show that $g$ has only finitely ...