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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586 views

How to prove that composition of two conformal functions is conformal

Let $\hat{\mathbb{C}}$ =$\mathbb{C}\cup\{\infty\}$. A theorem from my lecture notes says that a function $f: \hat{\mathbb{C}} \rightarrow \hat{\mathbb{C}}$ is conformal iff f is a linear fractional ...
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2answers
226 views

Find image under map of complex numbers

I need to find the image of the square $S={z; 0\le Re(z)\le1; 1\le Im(z)\le2} $under the function $f(z)=e^{ipz}$ where z is a complex number. I'm not sure what the variable p represents. Could it be ...
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1answer
59 views

Function differentiable in the complex sense

Let $G$ be a connected open, contained in $\mathbb{C}$, $f=u+iv$ a differentiable function in the complex sense in $G$ and $\bar{v}:\mathbb{C} \rightarrow \mathbb{R}$ a constant function. Prove that ...
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1answer
106 views

Showing that $\left| \int_{\gamma} \frac{dz}{z^2+1} \right| \leq \frac{\pi}{3}$

Here's a complex analysis question I'm fighting with. Let $\gamma$ be the arc of the circle $|z|=2$ that lies in the first quadrant. Show that $$\left| \int_{\gamma} \frac{dz}{z^2+1} \right| \leq ...
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1answer
122 views

How to calculate this complex integral $\int_0^\infty \frac{1}{q+i}e^{-(q+b)^2}\text{d}q$? (Please Help)

I want to carry out the following integration $$\int_0^\infty \frac{1}{q+i}e^{-(q+b)^2}\text{d}q$$ which is trivial if calculated numerically with any value for b. But I really need to get an ...
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1answer
169 views

Integrals using Cauchy integration formula

Determine, using Cauchy Integral formula, the value of $\int_{0}^{2 \pi}\frac{1}{2+\sin\theta}d\theta$ Progress; Substitute $z=e^{i\theta}$ and use that $\sin\theta=\frac{1}{2i}(z+\frac{1}{z})$ ...
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1answer
174 views

Liouville Theorem or Maximum Modulus Principle

$f$ and $g$ are two analytic functions on set $\Bbb C$ of all complex numbers such that $$f\left(\frac{1}{n}\right)=g\left(\frac{1}{n}\right)$$ for $n=1,2,3...$, then show that $f(z)=g(z)$ for each ...
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2answers
64 views

Does $Re\left( \int_{\gamma} f \right) = \int_{\gamma} Re(f)$?

I'm currently studying complex analysis. My current thinking is as follows: Let $f(t)=x(t)+iy(t)$. By definition, $$\int_{\gamma} f(t) \, \mathrm{d}t = \int_{a}^{b} f(\gamma(t)) \gamma'(t) \, ...
4
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1answer
115 views

Does a logarithmic branch point imply logarithmic behavior?

The complex logarithm $L(z)$ is given by $$L(z)=\ln(r)+i\theta$$ where $z=re^{i\theta}$ and $\ln(x)$ is the real natural logarithm. It is well known that $L(z)$ then sends each $z$ to infinitely many ...
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1answer
117 views

A Problem on Liouville's Theorem

Find all entire function $f$ such that $\vert f^{'}(z)\vert\leq M(1+\sqrt{\vert z\vert})$, for $M>0$. I did as follows: Let $f^{'}(z)=F(z)=\sum\limits^{\infty}_{n=0}a_nz^{n}$, $a_n=\dfrac{1}{2\pi ...
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1answer
65 views

analytic in open disc and $|f| = 1$ on boundary

$f$ is a function such that it is analytic in an open disc and on the boundary of the disc absolute value of $f$ is $1$. Does such function exist?
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3answers
101 views

How many distinct holomorphic function from $\mathbb{C}$ to $\mathbb{C}$ is there?

Is this proof OK ? Considering that the constant function $f(z) = c$ is analytic for all $c \in \mathbb{C}$, there is at least $\mathfrak{c}$ (the power of the continuum) holomorphic function from ...
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3answers
319 views

$|f|$ constant implies $f$ constant?

If $f$ is an analytic function on a domain $D$ and $|f|=C$ is constant on $D$ why does this imply that $f$ is constant on $D$? Why is the codomain of $f$ not the circle of radius $\sqrt{C}$?
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2answers
74 views

Change of variable x=iy in improper integral

I'm trying to solve the following question: Let $I=\int_0^{\infty}\exp(-x^4)dx$. Take $x=i y$ to get $i\int_0^{\infty}\exp(-y^4)dy=i I$. Explain. This change of variable implies $I=iI$, with $I$ ...
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0answers
62 views

Blaschke Product

I really need to understand Blaschke product very well. I have learned the basic material from Garnett's books on Bounded analytic functions and from Rudin's Real and complex analysis. But I'm looking ...
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0answers
58 views

Regarding Logarithmic of complex numbers

$\newcommand{\Log}{\operatorname{Log}}$ In my undergraduate complex analysis textbook, it claims that $$\Log(1+i)^2=2\Log(1+i)$$ I am not sure if this is a misprint or there is actually a way of ...
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1answer
102 views

Determining all possible values of contour integral of $\int\exp{z^{-1}}dz$

In our class, we are asked to find all possible values of $\int_{\gamma}\exp{z^{-1}}dz$ where $\gamma$ is any closed curve not passing through $z=0$. I wanted to ask if I can rewrite this expression ...
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1answer
284 views

Behavior of complex exponential as imaginery coefficient tends to infinity

This has occurred in my undergraduate complex analysis textbook: suppose $e^z=e^xe^{iy}$ , what happens when $y\Rightarrow\infty$ my intuition is that since $y$ denotes the phase of the circle with ...
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1answer
266 views

Show a certain subset of $\mathcal{O}(\mathbb{H})$ is a normal family

I need to show that $\mathcal{F}=\{f\in \mathcal{O}(\mathbb{H}): |f(z)|\neq 5 \forall z \in \mathbb{H}\}$ is a normal family. Here $\mathbb{H}$ is the upper half plane and $\mathcal{O}(\mathbb{H})$ is ...
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0answers
175 views

The family $\{ f(kz)\}_{k \in \mathbb C}$ with $f$ entire is normal in the annulus $r_1<|z|<r_2$ iff $f$ is a polynomial

As mentioned in the title, given $0 \leq r_1<r_2 \leq +\infty$, and an entire function $f : \mathbb C \to \mathbb C$, I want to prove that $$\mathfrak F= \{f(kz) \}_{k \in \mathbb C} \text{ is ...
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2answers
98 views

Show that the ring of holomorphic functions on the unit disc is not a local ring

I'm asked to show that the ring of holomorphic functions on the unit disc $\{z \in \mathbb{C} \mid |z| < 1\}$ is not a local ring. I'm quite sure that this is not a difficult proof, and I've ...
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2answers
59 views

Holomorphic functions with conditions

$f$ and $g$ are holomorphic on the same domain $D$. If $f(z)=u(z)+iv(z)$ and $g(z)=u(z)-iv(z)$, and $f(1+i)=2+3i$, then $g(4+3i)=?$ Is it possible to find $g?$
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2answers
109 views

Complex contour integrals

So I'm working through some practice questions for my complex analysis unit and I've come across this integral: $$\int_\Gamma {(3z^2-z)\over(z-1)^2(z+1)}dz$$ where $\Gamma$ is the following contour ...
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3answers
63 views

Writing $ e ^{3 z} $ in standard form.

I began by having: $z = x + iy$. Making $e^{3z} = e^{3x + 3iy}$ I get stuck here. I'm supposed to write this function as $ w = u(x,y) + iv(x,y)$
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0answers
46 views

Definite integral containing Bessel functions

I am stuck in calculating the integral: $$\int_0^\infty J_0(kr)J_1(kr_0)\exp\left(-it\sqrt{gk}\right)dk$$ Is it possible to use residue theorem? Thanks in advance.
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2answers
441 views

Finding the singularity type at $z=0$ of $\frac{1}{\cos(\frac{1}{z})}$

I have the following homework problem: What kind of singular point does the function $\frac{1}{\cos(\frac{1}{z})}$ have at $z=0$ ? What I tried: We note (visually) that $z_{0}$ is the same ...
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1answer
39 views

Writing $ h(z) =\frac{ z + i}{z^2 +1} $ in standard form

My attempt: $ h(z) =\frac{ z + i}{z^2 +1}$ Let $z = x + iy$ Then $$ \frac{ z + i}{z^2 +1} = \frac{x + iy + i}{x^2 - y^2 + 2ixy + 1}$$ This is approximately where I get stuck. I'm supposed to ...
2
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1answer
70 views

Confused about the explicit formula for $\psi_0(x)$

In the explicit formula for $\psi_0(x)$ used in the PNT proof : $$\psi_0(x) = x - \sum_{\rho} \frac{x^{\rho}}{\rho} - \frac{\zeta'(0)}{\zeta(0)} - \frac{1}{2} \log (1-x^{-2}) $$ In particular the ...
0
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1answer
121 views

Is there only one analytic continuation of the Riemann zeta function?

If I were to manipulate the zeta function in a 'new way' would I end up with an analytic continuation that is equal to the one know or something completely new for values less than 1 and complex ...
2
votes
1answer
135 views

How to find the inverse mellin transform?

On the wikipedia page http://en.wikipedia.org/wiki/Mellin_transform the second formula is an integral transformation for the inverse Mellin transform. Being new to integral transforms, I wonder how ...
2
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0answers
37 views

What's the difference between these two assumptions?

I see that the Cauchy Integral Formula follows from the Cauchy-Green Formula (both stated below), but I am unsure what the difference is in their assumptions. $\underline{Cauchy-Green \, Formula:}$ ...
27
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1answer
628 views

Proving $\sum_{n=-\infty}^\infty e^{-\pi n^2} = \frac{\sqrt[4] \pi}{\Gamma\left(\frac 3 4\right)}$

Wikipedia informs me that $$S = \vartheta(0;i)=\sum_{n=-\infty}^\infty e^{-\pi n^2} = \frac{\sqrt[4] \pi}{\Gamma\left(\frac 3 4\right)}$$ I tried considering $f(x,n) = e^{-x n^2}$ so that its ...
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0answers
42 views

The integral equation $\int_{-\infty}^{\infty}f(x) + af(g(x)) dx = b$

How to solve for $f(z)$ in the equation $\int_{-\infty}^{\infty}f(x) + af(g(x)) dx = b$ where 1) $f(x),g(x)$ are holomorphic near the real line. 2) $x$ is considered real here. 3) $a$ is a given ...
4
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1answer
159 views

Why holomorphic injection on $C^n$must be biholomorphic?

This result is certainly right in the 1-dim'l case. But I don't know how to show the general case by induction. Can anyone tell me the detail please?
2
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1answer
191 views

Using Cauchy-Riemann equations and continuity of the partial derivatives to decide if $f(z)$ is analytic

I'm doing an introductory course to complex analysis, and I'm having some trouble getting an overview on when a function is analytic and when it is not. I have a function $f(z)$ which is defined for ...
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1answer
147 views

Finding all u, v such that a function is holomorphic

2 similar questions Find all possible $u(t)$ and $v(x,y)$ for which the function $f(z) = u(xy) + iv(x, y)$ is holomorphic. Let $f$ be an entire function (analytical in $\mathbb{C}$) of the form ...
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1answer
144 views

Questions regarding the Riemann-Siegel $\theta$ Function

My questions are a request, please, for help in understanding some comments in the wikipedia article discussing the Riemann-Siegel $\theta$ function ...
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1answer
66 views

Circumventing Jordan's Lemma

Let $C_R$ be the semi-circle of radius $R$ in the upper half plane, centered at the origin (oriented counter-clockwise). I would like to prove that $$ \lim_{R\to\infty} \int_{C_R} \frac{e^{iz}}{z} dz ...
2
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2answers
96 views

Prove that $\lim_{n \to \infty}n\left(\frac{1+i}{2}\right)^n = 0$

I understand for this proof I must use the principle that I must find $N$, such that for $|u_n - 0| < \epsilon$ I have $n > N$ for $N$ that depends on $\epsilon$. However, I can't seem to get ...
2
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1answer
90 views

$z\mapsto\sin (\overline{z})$ is not holomorphic

I have to prove that the function $f:\mathbb{C}\to\mathbb{C}$ defined by $f(z)=\sin(\overline{z})$ is not holomorphic at any point of $\mathbb{C}$. Now, I want to show that $f$ does not satisfy the ...
2
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0answers
240 views

Using the theorem of Rouché in order to show the fundamental theorem of algebra

Infer from the theorem of Rouché that every non-constant polynomial does have a zero point in $\mathbb{C}$ (Fundamental Theorem of Algebra). Good day, consider the polynomial $$ ...
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1answer
57 views

Let $f(z) = \frac{2z-1}{3z+2}$. Prove that $ \lim_{h \to 0}\frac{f(z_0+h)-f(z_0)}{h} = \frac{7}{(3z_0+2)^2}$

I'm having a hard time with the problem stated. I understand this is an epsilon-delta proof. However, When I get to simplifying the numerator, I get $\frac{7h}{9z(z+h)+6(2z+h)+4}$ (this is through ...
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6answers
1k views

Solve $z^4+1=0$ algebraically

I know the result and how to solve it using trigonometry and De Moivre. However, given that the complex number $z$ can be rewritten as $a+bi$, how can I solve it algebraically?
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1answer
79 views

Help Sketching graph in $\mathbb{C}$

I'm recently reviewing complex analysis and I have trouble trying to sketch graphs involving $\arg z$. I want to graph the following set: $$\{z \in \Bbb C : |z|\le \arg z \text{ and } 0\le \arg ...
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1answer
85 views

Branch of a square root

Denote $\sqrt{}$ a branch of a square root. I want to show that the identity $\sqrt{\dfrac{1}{z}}=\dfrac{1}{\sqrt{z}}$ can be false, but I don't find a counterexample. Does anyone know a ...
2
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1answer
142 views

If $f(z)g(z) = 0$ for every $z$, then $f(z) = 0$ or $g(z) = 0$ for every $z$.

This is for homework, and I would really appreciate a hint. The question states "If $f$ and $g$ are holomorphic on some domain $\Omega$ and $f(z)g(z) = 0$ for every $z \in \Omega$, then $f(z) = 0$ ...
3
votes
1answer
168 views

A Cauchy principal value integral, using contour integration and Plemjel.

I came across the following integral $$lim_{\epsilon->0+}\int_\mathbb{R}\frac{e^{-ax^2+ibx}}{x+i\epsilon}dx$$ with a,b>0. Using Plemjel's formula led me to evaluating ...
2
votes
1answer
148 views

Show that $\sigma^{-\lambda(x) - 1}$ is continuous on $(0,1)$.

Let $V$ be an open connected subset of $\mathbb{C}$ and $A(V)$ be the set of all (complex-valued) analytic functions on $V$. If $\lambda \in A(V)$ with $\Re \lambda(x) < 0$ for all $x \in V$ , ...
2
votes
1answer
97 views

Diameter of the image of holomorphic $f: \mathbb{D} \to \mathbb{C}$

NOTE THE FOLLOWING QUESTION ATTEMPTS TO PROVE A FALSE STATEMENT. I LEFT IT UP THOUGH IN ORDER TO HELP ANYONE ELSE MAKING THE SAME MISTAKE AND TO GIVE CREDIT TO THE RESPONDENTS. Suppose I have a ...
2
votes
1answer
81 views

$\mathbb C\cup\{\infty\}$ is compact, a “direct proof”.

Consider the Riemann sphere $\mathbb C\cup\{\infty\}$ equipped with the usual topology. In most textbooks the compactness of $\mathbb C\cup\{\infty\}$ is proven by showing an explicit homeomorphism ...