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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show that for every $a \in D$ is a function $K_a \in H ^ 2 (D)$ such that $f (a)=\langle f, K_a \rangle_{H^2(D)}$.

Consider the Hardy space $H ^ 2 (D)$, where D is an open set in the complex. Show that$ H ^ 2 (D)$ is a Hilbert space, and show that for every $a \in D$ is a function $K_a \in H ^ 2 (D)$ such that $$f ...
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1answer
477 views

Prove that a bounded analytic function in the right half-plane which vanishes at each positive integer is identically zero.

I want to solve the example below, but I can not. Prove that a bounded analytic function in the right half-plane which vanishes at each positive integer is identically zero. Please, you will be ...
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1answer
93 views

image of the st. line $y=mx+c$

What is the slope of the image of the st. line $y=mx+c$ under the mapping $f(z)=az+b,a,b\in\mathbb C?$ I think I don't need the constant $b$ for translation don't rotates any shape. Letting $A$ ...
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1answer
145 views

Contour integral of $\displaystyle\int_\gamma \dfrac{1}{(2z+1)(z+3)^2}$

Im a little confused by the following integral question Let $\gamma$ be the unit circle in $\mathbb{C}$ traversed in the anti-clockwise direction. $\displaystyle\int_\gamma ...
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1answer
62 views

Justification in change of variables

it would be fantastic if anyone could help me with the following problem: I have the integral $$\operatorname{Im} \left( \int^\infty_0 e^{it} t^{s-1} \mathrm{d} t\right)$$ and I wish to make the ...
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2answers
746 views

Cauchy integral formula for $\displaystyle\int_\gamma \dfrac{\sin z }{z^4-16}dz$

I have working through past exam questions and I think I have the hang of the Cauchy integral formula and the extended formula... but am a little stuck with how to work these examples out... and the ...
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1answer
159 views

Rouché's Theorem on $z^{10} + 10z + 9$

Please note: this question was asked before, but the solution provided does not work as far as I know; see How to find the number of roots using Rouche theorem? We have $f(z) = z^{10} + 10z + 9$ and ...
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1answer
53 views

Prove that the given sum is not Fourier series

Book browsing Banach spaces of Analytic function of the author Kenneth Hoffman on page 74 is one example. This example is compiled in this way: Prove that $$ \sum_{n=1}^{\infty}\frac{1}{\log ...
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1answer
151 views

sufficiency and necessity of convergence of $\sum a_n$ wrt convergence of $\prod (1 + a_n)$

Does there exist a sequence $a_n$ of complex numbers such that $\sum _{i = 0}^\infty a_n$ converges and the product $\prod _{i = 0}^\infty (1+a_n)$ does not converge to any complex number(not even ...
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1answer
294 views

If $f$ is entire and $g$ has an essential singularity, must $f\circ g$ have an essential singularity?

I was trying to prove: Let $f:\mathbb{C}\to\mathbb{C}$ be entire. If $f(f(z))$ is a polynomial, prove that $f$ is polynomial. Assuming the contrary, we would get that $f(1/z)$ has an essential ...
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1answer
71 views

What type of singulatity does $f(z)=\exp\left(\dfrac{z}{1-\cos z}\right)$ has at $z=0?$

What type of singulatity does $f(z)=\exp\left(\dfrac{z}{1-\cos z}\right)$ has at $z=0?$ I'm completely clueless. Added: In $\mathbb C-\{0\},f(z)=\exp\left(\dfrac{z}{1-\cos ...
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2answers
322 views

Analytic $f$ with $|f|$ constant on $\partial D$, I need to show that it has atleast one $0$ inside $D$

Suppose that $f$ is analytic in a domain $G$ in the complex plane and not constant. Let $D$ be a disc whose closure is contained in $G$,$|f|$ constant on $\partial D$, I need to show that it has ...
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1answer
71 views

Is $f(2)=\dfrac{1}{4}?$

Let $f:\mathbb C\to\mathbb C$ be a meromorphic function analytic at $0$ satisfying $f\left(\dfrac{1}{n}\right)=\dfrac{n}{2n+1}~\forall~n\ge 1.$ Question: Is $f(2)=\dfrac{1}{4}?$ My attempt: By the ...
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1answer
116 views

How to prove this identity for ${}_3F_2$ (Generalized Hypergeometric Function)?

This may look like homework, but it is not. I've found this identity (using Mathematica): $$ {}_3F_2 \left( \matrix{1,1,1 \\ 2, e} ; 1 \right) = (e-1) \psi^{\prime}(e-1), $$ valid for $e$ with ...
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0answers
87 views

Which of the followings are correct: (Analytic Function)

Which of the followings are correct: My thought: $g(z)=\dfrac{2}{3+z}$ is analytic on $\mathbb C-\{3\}.$ Also $f=g$ on $\{\dfrac{1}{n}\}$ which has a limit point $0$ on $\mathbb D.$ So by identity ...
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4answers
55 views

Evaluating $\int_{0}^{2} \frac{t}{t+i} dt$

$$\int_{0}^{2} \frac{t}{t+i} dt$$ I have no idea how to even begin to split this up into real and imaginary parts. The only thing I can think of is that it might involve the use of natural log? Any ...
4
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3answers
388 views

Example of a Function with Infinitely Many zeros in the disc

One of the questions in my complex analysis book (Stein's text) is the following: Prove that if $f$ is holomorphic in the unit disc, bounded, and not identically zero, and ...
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3answers
55 views

Functions differentiable on {$z \in \mathbb{C}: 0 < |z| < 1$}

This is a past exam question from a Complex Analysis exam paper.. Prove or disprove that there exists a function $f$ differentiable on {$z \in \mathbb{C}: 0 <|z|<1$} such that $(i) ...
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2answers
3k views

Prove that the taylor series of cos(z) and sin(z) are holomorphic

I have a question on an old exam paper given as follows: If we define $$\cos(z) = 1 - \frac{z^2}{2!} + \frac{z^4}{4!} - ... \frac{z^{2n}}{(2n)!} + ... = \sum_{n=0}^{\infty} ...
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1answer
2k views

Finding poles and order of poles of functions

Im struggling with the concept of finding poles and then indicating their order.. The following past exam questions asks to find the poles and indicate their order ...
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3answers
274 views

Expanding $\frac{1}{1-z-z^2}$ to a power series.

How would you expand the analytic function $$\frac{1}{1-z-z^2}$$ to a series of the form $$\sum_{k=0}^\infty a_k z^k \, \, ?$$
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1answer
51 views

Uniqueness theorem for Rational Functions

I know that for polynomials $P,Q$, the equation $P(z) \equiv Q(z)$ implies that they are of the same degree and have the same coefficients. Is there an analogous result for rational fucntions? That ...
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1answer
64 views

Does it imply $|f(z)|\le |z|^n~\forall~z$ in the annulus?

$f$ is analytic in the annulus $1\le|z|\le R$ such that $|f(z)|\le1~\forall~|z|=1$ and $|f(z)|\le R^n~\forall~|z|=R.$ Does it imply $|f(z)|\le |z|^n~\forall~z$ in the annulus?
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102 views

prove an equation of complex numbers

How to prove this equation: $$\sin\left(\frac{\pi}{n}\right)\cdot \sin\left(\frac{2\pi}{n}\right) \cdots \sin\left(\frac{(n-1)\pi}{n}\right)=\frac{2n}{2^n}$$ There's a hint: Consider the product of ...
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3answers
113 views

Cauchy Riemann Equations for $h(z)=|z|^2z^2$

A past exam question I have come across is... Is the following function differentiable on $\mathbb{C}$ $h(z)=|z|^2z^2$ I have calculated $u(x,y)=x^4-y^4$ and $v(x,y)=2x^3y+2xy^3$ I then have ...
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0answers
30 views

Differentiability points of $f(z)=z^5(4\overline{z}+i|z|^2-(\text{Im }z)(\text{Re }z)^2)$

Find all points at which $f(z)=z^5(4\overline{z}+i|z|^2-(\text{Im }z)(\text{Re }z)^2)$ is differentiable. Of course, we can write $z=x+iy$ and check the Cauchy-Riemann equations. But this is just ...
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1answer
85 views

Can we conclude $f$ is analytic inside $C$?

Let $f$ be continuous on and inside a simple closed contour $C$ such that $\int_Cf(z)dz=0.$ Can we conclude $f$ is analytic inside $C?$
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1answer
119 views

Calculate partial derivatives of $exp(-(z^{-4}))$

I think I may be being stupid but I can't seem to work how to calculate the partial derivatives of $f(z) = exp(-(z^{-4}))$ at $0$. I understand that they should exist but you then use that this ...
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1answer
912 views

why holomorphic function $f$ can't extends continuously to its boundary such that $f(z)=1/z$?

Show that there is no holomorphic function $f$ in the unit disc $\Bbb{D}$ that extends continuously to $\partial \Bbb{D}$ such that $$f(z)=\frac{1}{z}$$ for $z \in \partial \Bbb{D}$. where ...
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2answers
73 views

Ascending sequence of simply connected sets and their union

I have a question about simply connected sets that might sound quite trivial and easy but i am not sure if my first step of the proof will be sufficient for the whole proof. So, we have an ascending ...
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1answer
82 views

radius of convergence of function defined by alternated addition and multiplication

Starting from $z\in\mathbb{C}$, define $r(z)$ starting from the initial value $v_{0}=0.0$ and repeat the following iteration from $1,2,\ldots$ $$v_{n+1} = \begin{cases} v_{n} + \frac{z^{n}}{n!} ...
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1answer
64 views

Does $f(z)=\sin\left(\dfrac{1}{z}\right)~\forall~z\in\mathbb C-\{0\}?$

Let $f$ be analytic in $\mathbb C-\{0\}$ such that $f\left(\dfrac{1}{n\pi}\right)=\sin\left(n\pi\right)~\forall ~n\in\mathbb Z.$ Does $f(z)=\sin\left(\dfrac{1}{z}\right)~\forall~z\in\mathbb ...
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0answers
135 views

What can we say about the entire functions that omit the value zero?

What can we say about the entire functions that omit the value zero? My thought: Since any nonconstant entire function comes arbitrarily close to any complex number the said functions are either ...
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1answer
92 views

Holomorphic function which is injective

Let $f$ be holomorphic in the upper half plane $\{\operatorname{Im} z> 0\}$. Suppose that $\operatorname{Im} f'(z)> 0$ for all $z$ in the half plane. Show that $f$ is 1-to-1.
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1k views

Zeta function zeros and analytic continuation

I'm learning about the zeta function and already discovered the intuitive proof of the Euler product and the Basel problem proof. I want to learn how to calculate the first zero of the Riemman Zeta ...
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2answers
82 views

Help with algebra involving residues

Evaluate $$\int_0^\infty \dfrac {\cos(mx)}{1+x^4}dx$$ I know I have to change the inegral to $-\infty$ as the lower limit. I understand the logic of the problem but it's the algebra in finding the ...
2
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2answers
206 views

Integration using residues

For the following problem from Brown and Churchill's Complex Variables, 8ed., section 84 Show that $$ \int_0^\infty\frac{\cos(ax) - \cos(bx)}{x^2} \mathrm{d}x= \frac{\pi}{2}(b-a)$$ where $a$ and ...
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2answers
172 views

Contour Integation $\int_\gamma \frac{\cos^2z}{z^2}$

I have the following question from a past exam paper that I'm not really sure how to evaluate. Any help would be appreciated... Let $\gamma$ be the unit circle in $\mathbb{C}$ traversed in the ...
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0answers
60 views

f analytic on unit disk implies f is real on (-1,1)?

Looking at a problem in Wunsch. Suppose f is analytic on the unit disk. Then I am asked to show that f must be real on the interval (-1,1). If f = u + i v, then how do I show that v(x,0) = 0 for ...
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1answer
242 views

whether or not there exist a non-constant entire function $f(z)$ satisfying the following conditions

In each of the case below, determine whether or not there exist a non-constant entire function $f(z)$ satisfying the following conditions. ($1$) $f(0)=e^{i\alpha}$ and $|f(z)|=1/2$ for all $z \in Bdr ...
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0answers
269 views

Complex Analysis and Proper Holomorphic Maps

Prove there is no proper holomorphic map from the unit disc into the complex plane. I know as $z$ approaches the boundary of the unit disc, $f(z)$ will approach infinity, hence unbounded. Can we ...
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1answer
45 views

How can two functions be compared?

If the derivatives of all orders for two functions agree at one point how can two functions be compared? There's no mention about the domain of those two functions. Can I suppose them to be ...
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1answer
144 views

How to determine where a function is complex differentiable

I know the definition of complex differentiability and also am aware that $f$ is complex differentiable at $z_0$ iff it is real differentiable at $z_0$ and that the partial derivatives satisfy the ...
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4answers
5k views

Radius of Convergence of power series of Complex Analysis

I have come across the following few questions on past exams papers.. I know how to solve these type when it is of the form $a_nz^n$ but don't have a clue what to do with these. Any help would be ...
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1answer
57 views

$\int_C\dfrac{f'(z)}{z-z_0}=\int_C\dfrac{f(z)}{(z-z_0)^2}$

$f$ is analytic everywhere on and inside a simple closed contour $C.$ Let $z_0$ be a point interior to $C.$ Then I've shown $\int_C\dfrac{f'(z) dz}{z-z_0}=\int_C\dfrac{f(z)dz}{(z-z_0)^2}$ as follows: ...
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1answer
107 views

What is the value of $\int_C\dfrac{f(z)}{z-z_0}dz?$

Suppose that $f$ is analytic inside and on a simple closed contour $C$,and $z_0$ lies outside $C.$ What is the value of $\int_C\dfrac{f(z)}{z-z_0}dz?$
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3answers
78 views

is $f$ analytic inside $C?$

If $$f(z_0)=\dfrac{1}{2\pi i}\int_C\dfrac{f(z)}{z-z_0}dz$$ for all point $z_0$ inside $C,$ is $f$ analytic inside $C?~(C:$ simple closed contour$)$
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1answer
173 views

Type of an entire function has uncountable zeros [duplicate]

If an entire function has uncountable zeroes then what can you say about the function?
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1answer
69 views

Uniqueness of singular measure for inner function

A singular inner function $M$ (an analytic function on the open unit disk without zeros which takes on unimodular boundary values almost everywhere) can be written as $$M(z)=c \exp\left(\int_0^{2\pi} ...
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1answer
49 views

Two complex functions whose real parts are equal vary only by a constant

Let $D$ be a bounded domain with boundary $B$. Suppose that $f$ and $g$ are both analytic on $D$ and continuous on $D \cup B$, and suppose further that $Re (f(z))=Re(g(z))$ for all $z\in B$. Show that ...