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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15
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2k views

What is the intuition behind the Wirtinger derivatives?

The Wirtinger differential operators are introduced in complex analysis to simplify differentiation in complex variables. Most textbooks introduce them as if it were a natural thing to do. However, I ...
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6answers
1k views

Natural derivation of the complex exponential function?

Bourbaki shows in a very natural way that every continuous group isomorphism of the additive reals to the positive multiplicative reals is determined by its value at $1$, and in fact, that every such ...
11
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3answers
1k views

An entire function is identically zero?

I'm preparing for a PhD prelim in Complex Analysis, and I encountered this question from an old PhD prelim: Suppose $f(z)$ is an entire function such that $|f(z)| \leq \log(1+|z|) \forall z$. Show ...
6
votes
2answers
465 views

Entire, $|f(z)|\le1+\sqrt{|z|}$ implies $f$ is constant

I am stuck on the following question. Given that $f$ is an entire function with $|f(z)|\le1+\sqrt{|z|}$ for all $z\in \mathbb{C}$, show that $f$ is constant. Can anyone give me a hint to get me ...
4
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3answers
466 views

erf(a+ib) error function separate into real and imaginary part

Is there an easy way to separate erf(a+ib) into real and imaginary part?
4
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1answer
1k views

$\lambda-z-e^{-z}=0$ has one solution in the right half plane

Let $\lambda > 1$ , want to show that the equation $$\lambda-z-e^{-z}=0$$ has exactly one solution in the right half plane $\{z:Re(z)>0\}$. Moreover, the solution must be real.I tried to use ...
4
votes
2answers
336 views

Branch point-what makes a closed loop around it special?

I am having difficulty understanding the concept of a branch point of a multifunction. It is typically explained as follows:branch point is a point such that the function is discontinuous when going ...
3
votes
2answers
142 views

What is a simple form of this integral?

This integral reminds me of something familiar but I cannot remember the rule to make it simple. $$\int_{-\infty}^{+\infty} \frac{\exp(i a \cdot v)}v \mathrm d v$$ where $a$ is a scalar for ...
2
votes
1answer
334 views

Calculating Riemann zeta function of a complex number given the complex contour integral

Can you please demonstrate how one would calculate the Riemann Zeta function of any complex number, given that the Riemann Zeta function is equal to the following (shown in ...
1
vote
3answers
230 views

Integral of $\log(\sin(x))$ using contour integrals

I know the integral is possible with a simple fourier series expansion of $-\log(\sin(x))$ But I am interested in complex analysis, so I want to try this. $$I = \int_{0}^{\pi} \log(\sin(x)) dx$$ ...
0
votes
0answers
94 views

Understanding Eigenvector

We have a matrix $A$ of size $N \times M$, where $N\le M$. Consider a vector $V$ of length $N$. Now I take product of $AV$ to get a vector $W$ of length $M$. Here I have projected the original ...
11
votes
2answers
580 views

Complex towers: $i^{i^{i^{…}}}$

If $w = z^{z^{z^{...}}}$ converges, we can determine its value by solving $w = z^{w}$, which leads to $w = -W(-\log z))/\log z$. To be specific here, let's use $u^v = \exp(v \log u)$ for complex $u$ ...
9
votes
1answer
474 views

Polynomial bounded real part of an entire function

Let $f(z)$ be an entire function whose real part is bounded by a polynomial in $|z|$. Does it follow that $f(z)$ is a polynomial? Or, without loss of generality and more suggestively ...
7
votes
2answers
1k views

What is the radius of convergence of $\displaystyle\sum z^{n!}$?

How could you find out the radius of convergence of $\displaystyle\sum z^{n!}$? I'm used to applying the ratio test to power series of the form $\displaystyle\sum a_{n}z^{n}$, but for a different ...
6
votes
1answer
263 views

Location of zeros of a sum of exponentials

Describe the approximate locations of the zeros of the function $$ f(z) = e^{iz}+e^{-iz}+e^z $$ lying outside the circle $|z|=R >>1$. Another prelim problem. For Rouche's theorem we need to ...
6
votes
2answers
796 views

All the zeroes of $p(z)$ lie inside the unit disk

Let $p(z) = c_0 + c_1z + c_2z^2 + \dots + c_nz^n$ where $0 \le c_0 \le c_1 \le \dots \le c_n$. I would like to show that all zeroes of this polynomial lie inside the unit disk by applying Rouche's ...
5
votes
3answers
257 views

Applications of the Residue Theorem to the Evaluation of Integrals and Sums

Evaluate the integral $$\int_{-\infty}^{\infty} \frac{1}{(1 + x^2)^{n+1}} dx. $$ I know that it equals $2\pi i$(the sum of the residues; at $z_k$) where $z_k$ are the poles of the function. I ...
5
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3answers
1k views

Using Residue theorem to evaluate $ \int_0^\pi \sin^{2n}\theta\, d\theta $

can you please guide me on evaluating this integral using residue theorem and binomial theorem $$ \int_0^\pi \sin^{2n}\theta\, d\theta $$ for $n = 1,2,3$ Honestly, I do not even know where to start, ...
5
votes
3answers
668 views

If $f^2$ and $f^3$ are analytic prove that $f$ is analytic at every point of $\mathbb{C}$. [duplicate]

Let $f : \mathbb{C} \to \mathbb{C}$ be continuous. If $f^2$ and $f^3$ are analytic prove that $f$ is analytic at every point of $\mathbb{C}$. if $f^2$ has no zero then $f=f^3/f^2$ and then it is ...
5
votes
1answer
187 views

sum of holomorphic functions

Does anyone know how prove the following? Suppose that $f,g$ are holomorphic functions on a non-empty open connected set $\Omega \subset \mathbb{C}$ and that $|f|^2+ |g|^2$ is constant on $\Omega$. ...
4
votes
1answer
250 views

Dirichlet problem in the disk: behavior of conjugate function, and the effect of discontinuities

Dirichlet's problem in the unit disk is to construct the harmonic function from the given continuous function on the boundary circle. It is solved by the convolution with the Poisson kernel, and we ...
4
votes
2answers
193 views

Evaluate these infinite products $\prod_{n\geq 2}(1-\frac{1}{n^3})$ and $\prod_{n\geq 1}(1+\frac{1}{n^3})$

What is $\prod\limits_{n\geq 2}(1-\frac{1}{n^3})=?$ $\prod\limits_{n\geq 1}(1+\frac{1}{n^3})=?$ I am sure about their convergence. But don't know about exact values. Know some bounds as well. For ...
4
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2answers
670 views

Proving surjectivity of $\cos(z)$ and $\sin(z)$ and find all $z : \cos(z) \in \mathbb R$ and all $z: \sin(z) \in \mathbb R$

I am trying to solve the following two problems: 1) Prove that the functions $\cos(z)$, $\sin(z)$ are surjective over the complex numbers. 2) Find all $z \in \mathbb C$: $cos(z) \in \mathbb R$ and ...
4
votes
2answers
167 views

Show that $f(z)=0$.

Suppose that $f:\mathbb{C}\rightarrow\mathbb{C}$ is analytic on the open unit disc and continuous on the closed unit disc. Assume that $f(z)=0$ on an arc of the circle $\{z\in\mathbb{C}:|z|=1\}$. Show ...
3
votes
2answers
147 views

The ratio $\frac{u(z_2)}{u(z_1)}$ for positive harmonic functions is uniformly bounded on compact sets

I want to prove the following: If $E$ is a compact set in a region $\Omega \subset \mathbb C$, prove that there exists a constant $M$, depending only on $E$ and $\Omega$, such that every positive ...
3
votes
4answers
925 views

Improper integral of $\sin^2(x)/x^2$ evaluated via residues

I have come across another improper integral I wish to evaluate via residues. The integral is: $$\int_{-\infty}^\infty{\frac{\sin(x)^2}{x^2}}dx$$ $\sin(z)$ behaves in an uneasy way so I tried ...
3
votes
3answers
630 views

How can I show this statement.

Show that there is no holomorphic fuction $f$ in the unit disc $D$ that extends continuously to boundary of $D$ such that $f(z)=\frac{1}{z} ~for~ z\in \partial( D) $. I tried to apply maximum ...
3
votes
3answers
1k views

Proof of Cauchy-Schwarz inequality - Why select s so that so that $||x-sy||$ would be minimized?

I was looking at a number of different proofs of the cauchy schwarz inequality in an inner product space ($\mathbb{R}^n$ or $\mathbb{C}^n$). All of them used the idea of $||x-sy||$ where $s$ was ...
3
votes
2answers
564 views

if a complex function $f$ is real-differentiable, then $f$ or $\overline{f}$ are complex-differentiable

This is an exercise from Remmert's Theory of Complex functions. Let $D\subset \mathbb{C}$ be a domain and $f:D\rightarrow \mathbb{C}$ a real-differentiable function. Assume that the following limit ...
2
votes
2answers
655 views

Show that $\sum_{n=0}^\infty r^n e^{i n \theta} = \frac{1- r\cos(\theta)+i r \sin(\theta)}{1+r^2-2r\cos(\theta)}$ [closed]

Show that $$\sum_{n=0}^\infty r^n e^{i n \theta} = \frac{1- r\cos(\theta)+i r \sin(\theta)}{1+r^2-2r\cos(\theta)},$$ where $0\leq r <1$. Using this, prove that $\sum_{n=0}^\infty r^n ...
2
votes
2answers
1k views

Find conformal mapping from sector to unit disc

Find a conformal mapping between the sector $\{z\in\mathbb{C} : -\pi/4<\arg(z) <\pi/4\}$ and the open unit disc $D$. I know that it should be a Möbius transformation, but other than that I am ...
2
votes
1answer
165 views

Prove the following equation of complex power series.

Show that for $|z| \lt 1$ with $z \in \Bbb C$, we have $$ \sum_0^\infty \frac{{z^2}^k}{1-{z^2}^{k+1}} = \frac{z}{1-z} $$ $$ \sum_0^\infty \frac{2^k{z^2}^k}{1+{z^2}^{k}} = \frac{z}{1-z} $$ My guess ...
1
vote
1answer
154 views

Finding the residue of function with Laurent series $\sum_{n=0}^{\infty}\sum_{k=0}^{\infty}\frac{y^n(A+By+Cy^{-1})^k}{\beta (\beta i)^n \ k!}$

I have been trying to find the residue of $f(\omega) = \frac{e^{i \omega a} e^{\frac{-b \omega}{\omega + ib}}}{i \omega}$ at the essential singularity $\omega = -ib$ for a while, but it is giving me ...
1
vote
1answer
485 views

Integrating squared absolute value of a complex sequence

I was reading through my book in complex analysis and i encountered this problem. Given, $F=\sum_{n=0}^{\infty} a_nX^n$ is a convergent power series with radius of convergence R. We are asked to show ...
1
vote
2answers
711 views

How to prove error function $\mbox{erf}$ is entire (i.e., analytic everywhere)?

How do I prove the error function $$ \mbox{erf}(z) = \frac{2}{\sqrt{\pi}} \int_{0}^{z} e^{-t^{2}} dt. $$ is entire? Could you give me some scratch proof?
1
vote
1answer
270 views

Representation of Holomorphic Functions By Exponential

Let $f$ be holomorphic and nonzero on $D_{1}(0)$ the open unit disc. Can we write (for the given domain) $f(z) = e^{h(z)}$ where $h$ is holomorphic? This seems clear using a naive log argument but I'm ...
1
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2answers
474 views

Area of Validity of Writing an Exponential Integral as Sum of IntegralSinus and -Cosinus

I'm confused by the two online references shown below. To me, they give different areas of validity of writing an exponential integral as sum of integralsinus and -cosinus. On this Wiki page, I find ...
0
votes
1answer
105 views

Lagrange Bürmann Inversion Series Example

I am trying to understand how one applies Lagrange Bürmann Inversion to solve an implicit equation in real variables(given that the equation satisfies the needed conditions). I have tried looking for ...
7
votes
1answer
557 views

Does there exist an holomorphic function such that $|f(z)|\geq \frac{1}{\sqrt|z|}$?

I have some trouble solving this problem: Does there exist an holomorphic function $f$ on $\mathbb C\setminus \{0\}$ such that $|f(z)|\geq \frac{1}{\sqrt|z|}$ for all $z\in\mathbb C \setminus ...
6
votes
4answers
245 views

What did Johann Bernoulli wrong in his proof of $\ln z=\ln (-z)$?

Some people say, Johann Bernoulli has proven $\ln z=\ln (-z)$ in the following way $$\ln ((-z)^2 )=\ln(z^2)\;\;\;\Rightarrow\;\;\;2\ln(-z)=2\ln z\;\;\;\Rightarrow\;\;\;\ln (-z)=\ln z$$ While the ...
6
votes
3answers
5k views

Mapping half-plane to unit disk?

Say you have the half-plane $\{z\in\mathbb{C}:\Re(z)>0\}$. Is there a rigorous explanation why the transformation $w=\dfrac{z-1}{z+1}$ maps the half plane onto $|w|<1$?
6
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1answer
458 views

3 holomorphic functions, sum of absolute values does not have maximum

I have the following problem: Let $f,g,h$ be holomorphic functions (non-constant) in some domain $D$. Show that the function $F(z):=|f(z)|+|g(z)|+|h(z)|$ has no local maximum in this domain $D$. ...
6
votes
2answers
438 views

How to compute the infinite tower of the complex number $i$, that is$ ^{\infty}i$

Let $x = i^{i^{i^{i^{.^{.^{.{^ \infty}}}}}}}$. This is the solution of the equation $i^x - x = 0 $ . I used Euler's identity to find a solution. But I haven't yet found the real and imaginary parts of ...
5
votes
1answer
661 views

Show that an entire function bounded by $|z|^{10/3}$ is cubic

Question: Let $f$ be an entire function such that $|f(z)|\leq1+2|z|^{10/3}$ for all z. Prove that $f$ is a cubic polynomial Thoughts so far: Using a corollary of Liouville's theorem, we know that we ...
5
votes
2answers
226 views

Domination of complex-value polynomial by highest power.

The problem: Let $P(n)$ be a polynomial of degree $n$. Let $$M(r):= \underset{|z|\le r}{\mbox{sup}} \hspace{2mm} \left|P(z)\right|.$$ I desire to establish that $$r\mapsto \frac{M(r)}{r^n}$$ for ...
4
votes
3answers
1k views

Complex Integration poles real axis

In class my professor said that $$ \int_{-\infty}^{\infty}\frac{e^{iax}}{x^2 - b^2}dx = -\frac{2\pi}{b}\sin(ab) $$ where $a,b > 0$. However, since the poles are on the real axis, isn't the integral ...
4
votes
2answers
1k views

If $|f|$ is constant, so is $f$ for $f$ analytic on a domain $D$.

I'm reading through a development of the maximum modulous principle, but I am stuck verifying a remark: $$\text{"it is enough to show that $|f|$ is constant, from which we may conclude that $f$ ...
4
votes
5answers
361 views

Fourier transform of $f(x)=\frac{1}{x^2+6x+13}$

How to find the Fourier transform of the following function: $$f(x)=\frac{1}{x^2+6x+13}$$
4
votes
1answer
416 views

An entire function with two periods

Can anybody help me with this question: If $f(z)$ is an entire periodic function and it has to periods $2$ and $2i$, how can I find all other periods?
3
votes
1answer
127 views

convergence of a particular series

Let be $ \Lambda\subseteq \mathbb C$ a lattice, I don't understand why the series $$\sum_{\lambda\in\Lambda\setminus\{0\}} \frac{1}{|\lambda|^s}$$ converges for $s>2$. Can someone help me?