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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24 views

Applying Cauchy's theorem

Why is the part highlighted in green equal to zero?
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15 views

Prove $SU(2)$ is isomorphic to the group of quaternions of norm 1

How could I start finding the isomorphism? Intuitively, a quaternion can be expressed as two complex numbers $a+bi+cj+dk=a+bi+(c+di)j$, and as an element of $SU(2)$ is $\left[ \begin{array}{ c c } ...
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11 views

Complex Vector spaces inner product superposition axiom

In my studies of Quantum mechanics, the following problem with complex vector spaces has come up, specifically as regards the inner product in such a space. Now in Shankars "Principles of Quantum ...
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11 views

Complex Differentiability and its difference to real differentiability

I am currently studying a course on complex analysis and complex differentiability is defined as: $f : U \rightarrow \mathbb{C}$, where $U$ is a domain, is complex differentiable if and only if it is ...
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3answers
141 views

Does $\sqrt{i + \sqrt{i+ \sqrt{i + \sqrt{i + \cdots}}}}$ have a closed form?

I've been brushing up on my complex analysis recently, and I've come across a problem that's stumped me: What are the real and imaginary parts of $$\sqrt{i+\sqrt{i+\sqrt{i+\sqrt{i+\cdots}}}} ?$$ I ...
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15 views

Analytic inverse of $f, f(0) = 0, f'(z) \neq 0 $ within minimum modulus on boundary.

Suppose $f(z)$ is analytic on open disk of radius $r$ and $f(0)=0$, $f'(z) \neq 0$. Show that $f$ has an analytic inverse on $\{|z| \leq m\}$ where $m$ is the minimum of $|f(z)|$ on $\{|z| = r\}$. And ...
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27 views

Question about a challenging problem in complex analysis, any ideas are welcome [on hold]

Is there any body who have any idea to solve this problem? It seems that it is a challenging problem. The problem is as follows:
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29 views

True or False: If f is analytic and maps the deleted unit disk to the unit disk, then 0 is not a pole for f.

I am studying for my final exam in complex variables and I ran across this true or false question. True or False: If $f:D(0;1)\setminus\{0\} \rightarrow D(0;1)$ is analytic, then $0$ is not a pole ...
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1answer
21 views

Theorem regarding primitives and complex integration

In order to apply this theorem must $f$ be continuous in the entirety of the open set $\Omega$ or only on the curve $\gamma$?
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25 views

Complex singularity exponent

I am studying about complex singularity exponents of holomorphic functions. I need some help to clarify a few things: First, what a complex singularity exponent is, for the holomorphic function ...
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1answer
37 views

Computing the complex integral?

I am dealing with the following: $$\int_{0}^{\infty}\frac{x\sin(x)}{(z^2+a^2)(z^2+b^2)}dx$$ Furthermore, I know $a,b>0$ and I know $a\neq b$. I believe this is using Jordan's Lemma? I see that the ...
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23 views

Integrating $\int_0^1\cos(\lambda x^3)dx$ using the saddle point method [on hold]

Find the leading term of asymptotics as $\lambda\to\infty$ $I(\lambda)=\int_0^1\cos(\lambda x^3)dx$ Using method of saddle points along a certain contour. I am having trouble approaching this ...
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15 views

How to determine contours by looking at the exponential integrands?

I know that we determine the contours in contour integrals by looking at the exponential integrand (assuming there is indeed an exponential integrand in the given integral) but I don't know how. For ...
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29 views

How are singularities of complex functions classified?

I have the function $$\frac{z^2+1}{z^3+6z^2+z}$$ And I wish to find the residues at $z_0=0$ and $z_0=2\sqrt{2}-3$ because they are within my given contour. However, I am really confused when it comes ...
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2answers
28 views

Have I Correctly Defined the Set of Nonzero Complex Numbers $\mathbb{C^*}$?

If the set of complex numbers $\mathbb{C} = \{a+bi\mid a,b \in \mathbb{R}\}$, then what would be the definition of the set of nonzero complex numbers? Am I right in defining such a set as ...
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12 views

Is the Möbius inversion applicable in the case of number functions with values in $Q(x)$

I am looking for the cause of an erroneous calculation I did the details I cant present here. I guess a "Möbius inversion" I apply might be the cause. Normally the Möbius inversion is valid for ...
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2answers
40 views

Image of curves in the complex plane

I'm not really sure what I'm being asked in this question. If $x=C,y=C$ doesn't that mean $z=C+iC$?
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1answer
14 views

Showing a complex function is nowhere differentiable in a certain disc

I have a function and I am asked to prove that it is nowhere differentiable on an open disc. I found the cauchy riemann equations and saw that is is satisfied at the origin. I don't know what to do ...
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0answers
29 views

Counting zeros of an analytic function [on hold]

Suppose f is analytic on $Ball_R(0)$ and satisfies $|f (z)| < R$ for $| z| = R$. Using Complex analytic methods (such as Rouche's theorem), how can I find the number of solutions (counting ...
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1answer
31 views

Help With Bromwich Inversion Formula Proof

To prove(copied from handwritten notes so possibly wrong): Bromwich Inversion Formula. Fix $x_0∈ℝ $. If $F$ is complex analytic on $\{z:\Re z > x_0\} $ and for every $x>x_0$, $y↦ F(x + iy )$ ...
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2answers
36 views

Under what conditions do you use that $\operatorname{Res}{(f(z)/g(z))}=f(z_0)/g'(z_0)$?

In complex analysis, this seems to be a really helpful way to avoid having to expand out Laurent series. I am unclear, however, when it is appropriate to use this property. In specific, I'm worried I ...
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13 views

Laurent series expansion for powers of n?

I wish to expand the function: $$\dfrac{e^z}{z^n-c^n}$$ about the point $z_0=c$, where c is a constant greater than 0 and n is greater than 2. So I have that $e^{z-c}$ expands to ...
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1answer
28 views

Why is (-1)^(2/3) equal to -1/2+(i sqrt(3))/2

Can someone please explain to me how $(-1)^{\frac2 3}$ can be written as $\frac {-1}{2}+\frac{i \sqrt3} 2$ ? Do you use the corrolation $(-1)^c = e^{(i c \pi)}$, where ${c}$ is a constant?
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31 views

Zeros of polynomials with real exponents

Does every non-constant function of the form \begin{equation*} f(z)=a_0+a_1z^{r_1}+\ldots a_nz^{r_n} \end{equation*} have a complex zero? Here the $r_k$ are positive reals, the $a_k$ are arbitrary ...
2
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1answer
23 views

Double period except for poles

I'm trying to solve a problem in Complex Analysis whose function $f$ is defined in $\mathbb{C}$, is meromorphic and have double period $(f(z)=f(z+a)=f(z+b),\ \frac{a}{b} \notin \mathbb{R})$ except for ...
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2answers
23 views

Applying Cauchy Residue Theorem to $\int_{C}\frac{e^{z}}{sin^2{z} - 1}$

For $\int_{C}\frac{e^{z}}{sin^2{z} - 1}$, $C = \{|z|=3 \}$, this has singularities at $z = \frac{\pi}{2}$ and $z = \frac{3\pi}{2}$. So $Res(f,\frac{\pi}{2}) = \frac{e^{z}}{\sin(2z)} = ...
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1answer
29 views

Residues of $z^2\sin(\frac{1}{z})$

I must find the residues of $z^2\sin(\frac{1}{z})$ at $z = 0$. Since $z = 0$ seems to be an Essential Singularity, i'm not sure how I can continue to find the residue of the function. Usually I am ...
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1answer
22 views

Harmonic functions proof

I don't understand here why: $2(\Delta(u_x)^2+\Delta(u_y)^2) \geq 0$. Here $\Delta= \nabla^2, \quad u'_x=u_x $ etc
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4answers
30 views

Find Solution of trigonometric complex equation

Find the solutions of $\sin z = 3$ There are 2 ways to solve this, I know how to do this with: $\sin z = \frac{1}{2i}(e^{iz}-e^{-iz}) = 3$ Now, I am now doing in the way: $\sin z = \sin x \cosh y+i ...
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2answers
109 views

Dog Bone Contour Integral

Would someone please help me understand how to integrate $$ \ \int_0^1 (x^2-1)^{-1/2}dx\, ? $$ This is a homework problem from Marsden Basic Complex Analysis. The text book suggested using a "dog ...
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0answers
37 views

An entire function is zero

Suppose $f(z)$ is an entire function that have zero on positive integers. Does it follow $f$ is identically zero? This seems like an application of Liouville theorem. But I cant come with a function ...
2
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3answers
64 views

Is $(a+bi)(a-bi) = a^2 + b^2 $ solely a real number or a complex number?

I have not dealt with complex numbers for a while now, but I was wondering if I multiplied the complex number $a+bi$ by its conjugate $a-bi$ to obtain $$(a+bi)(a-bi) = a^2 + b^2 $$ where $a,b \in ...
2
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1answer
38 views

A problem about elliptic functions

I am trying to solve some problems in complex analysis, but I am not succeeding in the following problem. Suppose that $f$ is a function with the following properties: $f$ is non-constant; $f$ is ...
4
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1answer
58 views

A Funtional equation in Complex variables

I have been stuck on this problem for a long time : If $f(z)=u(x,y)+iv(x,y)$ , prove that a. $f(z)=2u(z/2,(-iz)/2) +$ constant b.$f(z)=2iv(z/2,(-iz)/2) +$ constant This result seems very ...
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2answers
35 views

$f=u+iv$ holomorphic, $xu+yv = (x^2+y^2)e^x \cos y$, what is $f$?

$f(z)=u(x,y)+iv(x,y)$ holomorphic, $xu+yv = (x^2+y^2)e^x \cos y$, what is $f$? I tried to interprete $xu+yv$ as some part of a new function, for example, as the real part of $\overline{z}f$,but this ...
2
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0answers
38 views

$\frac{df}{dz}$ and $\frac{\partial f}{\partial z}$

If $f(z)=u(x,y)+iv(x,y)$, $z=x+iy$ what is the difference between $\frac{df}{dz}$ and $\frac{\partial f}{\partial z}$? I understand $\frac{\partial f}{\partial z}=\frac{1}{2}(\frac{\partial ...
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8 views

Show that the preimage of any circle or line in $\mathbb{C}$ under the stereographic projection is a circle on $\mathbb{S}^2$.

So this is an exercise in my complex analysis notes, which has been given the solution as follows (where I write $z = π(x_1,x_2,x_3) = \displaystyle\frac{x_1}{(1-x_3)} + \frac{ix_2}{(1-x_3)}$ for the ...
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0answers
16 views

Holomorphicity and partial derivatives [on hold]

If $f(z)=u(x,y)+iv(x,y)$, $z=x+iy$ and $f(z)$ is holomorphic, why is it the case that $u_{xx}$, $u_{yy}$, $u_{xy}$ and $u_{yx}$ exist and $u_{xy}=u_{yx}$?
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1answer
17 views

Holomorphy of continuous function on $(A\cup B)^C$

Let $f:\mathbb C\to \mathbb C$ be a continuous function and $A,B\subseteq \mathbb C$ two open connected sets with $\overline{A \cup B}=\mathbb C$. Further, we know that $f\mid_A$ and $f\mid_B$ are ...
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2answers
32 views

calculate the principal part of $\tan(z)$ at $\frac{\pi}{2}$

calculate the principal part of $\tan(z)$ at $\frac{\pi}{2}$. of course $\tan(z) = \frac{\sin(z)}{\cos(z)}.$ Because $\cos(z)$ is of order 1 in $\frac{\pi}{2}$ we know that our primal part must look ...
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1answer
33 views

Explain a complex number identity

A college math instructor provided me with the following: $$\left|1+e^{ix}\right|^2=\left(1+\cos x\right)^2+\sin^2 x$$ Can anyone show me how this is done?
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1answer
19 views

Let $C=\partial D_1(\mathbf i/2)$, compute $\int_C\frac{dz}{z^2+1}$

Let $C=\partial D_1(\mathbf i/2)$, compute $\int_C\frac{dz}{z^2+1}$ $C=\partial D_1(\mathbf i/2)$ is the boundary of the disc with center $\mathbf i/2$ and radius $1$, then $\mathbf i$ is ...
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3answers
44 views

Show that $\sum_{n \neq 0} \frac{(-1)^{n+1}}{in} e^{in\theta} = 2 \sum_{n=1}^\infty (-1)^{n+1} \frac{\sin n\theta}{n}$.

Show that $$\sum_{n \neq 0} \frac{(-1)^{n+1}}{in} e^{in\theta} = 2 \sum_{n=1}^\infty (-1)^{n+1} \frac{\sin n\theta}{n}.$$ This is not an exercise. It is an example from Stein, Fourier Analysis ...
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0answers
27 views

Confusion between partial and straight derivative wrt z

If $f(z)=u(x,y)+iv(x,y)$, $z=x+iy$ under what conditions is $f_z=\frac{df}{dz}$?
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1answer
33 views

Evaluation of $\prod_{k=1}^{\infty}\frac{a+k^2}{b+k^2}$

While playing around with the question The convergence of a sequence with infinite products, I found Mathematica to give me the result $$ \prod\limits_{k=1}^{\infty}\frac{a+k^2}{b+k^2} = ...
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0answers
18 views

What is a bounded sequence of holomorphic functions?

Let $\Omega\subseteq\Bbb C^n$ open, $\{f_n\}_n\subseteq\operatorname{hol}(\Omega,\Bbb C)$ bounded. What does this mean? A numerical sequence $(a_n)_n\subset\Bbb C$ is bounded if $\exists M>0$ ...
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1answer
67 views

Prove that there is no function $f$ that is analytic. [duplicate]

Prove that there is no function $f$ that is analytic in $\mathbb{C}\setminus\{0\}$ and satisfies $$|f(z)|\geq\frac{1}{\sqrt{|z|}},\quad \operatorname{for all}\quad z\in\mathbb{C}\setminus\{0\}$$ I am ...
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2answers
44 views

How to show real analyticity without extending to complex plane

Suppose we have some $f \in C^\infty(\mathbb{R},\mathbb{R}).$ For example, $$f(x)=(1+x^2)^{-1}.$$ Using complex analysis, we can easily show $f$ is real analytic. Is there an easy, general method ...
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0answers
24 views

Cauchy's Integral Formula: Where have I gone wrong?

I have the function $$M(\mathbf{r})=\frac{\pi}{2}\left(erf\left(9-\lvert \mathbf{r} \rvert\right)+1\right)$$ where $erf(x)$ is the usual error function. Since $\mathbf{r}$ is $\in \mathbb{R^2}$, I ...
2
votes
2answers
40 views

Show that $|e^z -1| \leq e^{|z|}-1$ for any z

Show that $|e^z -1| \leq e^{|z|}-1$ What i have tried is Let $z=x+iy$.Then, $$|e^z-1|=|e^x\cos y-1+ie^x\sin y|=\sqrt{(e^x\cos y-1)^2+(e^x \sin y)^2}=\sqrt{e^{2x}-2e^x\cos y+1}$$ I stuck here and ...