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, ...

learn more… | top users | synonyms (2)

2
votes
1answer
18 views

Apply the Cauchy-Goursat theorem to show that $\int_C \operatorname{Log}(z+2)\, dz=0$ on a unit circle.

Cauchy-Goursat theorem. If a function $f$ is analytic at all points interior to and on a simple closed contour $C$, then $$\int_C f(z) \,dz=0.$$ This is a problem from Churchill's Complex Variables. ...
0
votes
1answer
25 views

Differentiability on $Re(z)^2$

Suppose that $l(z)=Re(z)^2$. Is $l$ differentiable at $z_0 \in \mathbb{C}$? What is $f'(z_0)$? Where is $l$ differentiable? Write $w=u+iv$ and $z_0=x_0 +iy_0$. So $$\lim \limits_{w \rightarrow 0} ...
0
votes
1answer
21 views

Find and classify singularities

Let $f(z)=\frac{e^{-z}\sin(2(z-1)^2)}{(z^2-4)(z-1)^2}$. Find and classify the singularities of $f(z)$. So far I have that there are singularities at $1$, $2$ and $-2$ $2$ and $-2$ are simple ...
0
votes
2answers
27 views

Find and classify the singularities of $f(z)$

Let $$f(z) = \frac{e^z\sin(3z)}{(z^2-2)z^2}$$ Find and classify the singularities of $f(z)$. So far I have that there are singularities at $0$, $\sqrt{2}$ and $-\sqrt{2}$. Are these correct? ...
0
votes
1answer
16 views

Boundary preserving map

Let $K\subseteq\mathbb{R}^2$ be a compact set. Is it true that for a continuous map $p:K\to\mathbb{R}^2$ we have: $p(\partial K)=\partial p(K)$? Are there any generalizations? P.S. Note that ...
0
votes
1answer
15 views

Runge's Theorem for meromrophic functions

Is there a name for this extension of Runge's theorem? Theorem: Let $K\subset\mathbb{C}$ be compact, and let $A\subset K^c$ be a set which intersects each component of $K^c$. Let $f$ be meromorphic ...
1
vote
0answers
25 views

Multivalued Functions for Dummies

I have been studying complex analysis for a while, but I still cannot "get" how multivalued functions work. Despite having it explained to me many times, my brain cannot process it. For example, I ...
0
votes
0answers
22 views

Complex limit proof

Definition: Say $f: S \longleftrightarrow \mathbb{C} \longleftrightarrow \mathbb{R}^2$, $z_0 \in S$, $l \in \mathbb{C}$. We say $$\lim \limits_{z \rightarrow z_0} f(z) = l_0$$ if $\forall \varepsilon ...
1
vote
1answer
72 views

Proving an entire function is constant

I'm trying to prove that the entire functions such that \begin{equation*} n^2f(1/n)^3+f(1/n)=0 \end{equation*} for all $n\in\mathbb{N}$, are constant. I suppose I should prove that $f$ is bounded ...
0
votes
1answer
16 views

Derivative of a complex conjugate

I anticipate that this is a stupid question, but suppose $c \in C$. What is $\frac{\partial c^{*}}{\partial c}$? I've been trying and failing for about an hour to figure it out from the definition of ...
0
votes
1answer
37 views

problems with singularity $0$ of $\int_{W} \frac{e^{\frac{1}{z}}}{(z-3)^3} dz$.

I have the complex integral \begin{equation*} \int_{W} \frac{e^{\frac{1}{z}}}{(z-3)^3} dz \end{equation*} on a circle with radius $6$. Obviously we have two singularities, one in $0$ and one in $3$. ...
3
votes
1answer
15 views

$\{\,z\in \mathbb C : \operatorname{Im}(z)>-1 , |z|<2\, \}$ onto upper half space $\{\,z\in \mathbb C : \operatorname{Im}(z)>0 \,\}$

I am search an one to one mapping that maps the domain $$\{\,z\in \mathbb C : \operatorname{Im}(z)>-1 , |z|<2\, \}$$ onto upper half space $$\{\,z\in \mathbb C : \operatorname{Im}(z)>0\, ...
0
votes
1answer
29 views

Existence of analytic function on a unit dsic (Converse of Schwaraz Pick Lemma )

(Schwarz - Pick Lemma) Suppose that $f$ is analytic on the Unit Disk $\triangle$ and satisfies the following two conditions : (1) $|f(z) \leq 1$ for all z $\in \triangle$ (2) $f (a) = b $ for some ...
1
vote
1answer
13 views

Contour integral of non analytic exponential function

The value of the integration of the function $f(z)$ over the circle of radius 3 centered at $z=1$, where $f(z) = e^{\frac{-1}{(z-1)^2}}$ this function has a pole at $z=1$ of $2$nd order. I don't ...
3
votes
2answers
74 views

Is entire function a polynomial? [duplicate]

Let $f:\mathbb{C}\to \mathbb{C}$ be an entire function, and suppose that for every $z\in \mathbb{C}$ there exists $n_z\in \mathbb{N}$ such that $f^{(n_z)}(z)=0$. Is $f$ necessarily a polynomial?
1
vote
1answer
11 views

Classification of conformally equivalent annuli via periods

How does one define the periods that appear in this question and show they are conformally equivalent? Are the details worked out in a textbook somewhere? Presumably we do something like take the ...
0
votes
1answer
17 views

Normal Family complex

I search about a family of holomorphic complex function that not normal but their derivative is normal. the definition of normality similar to Montel's Thrm
1
vote
1answer
31 views

Given $z$ show that $\left | z\right | = 2\sin\theta$ and $\arg z = \theta$

I've been attempting this complex-related question but couldn't quite crack the challenge. (b) Given that $$z={1-\cos 4\theta+i\sin 4\theta\over\sin 2\theta+2i\cos^2\theta},$$ show that ...
1
vote
1answer
16 views

Complex Analysis Triangular Inequality

I recently started learning Complex Analysis and as of now don't have much command over it, I am stuck up with this assignment Question of mine which is as follows: If $|Z_i| < 1$ and $V_i ≥ 0$ ...
1
vote
0answers
25 views

Complex Analysis Modulus Property

My teacher has asked an extra question apart from the one mentioned below in the picture: What will happen if |z| = |w| = 1 ? With respect to following answer: I don't see any point in that ...
1
vote
1answer
13 views

Orthogonal complex matrices: polar decomposition

Is there a decomposition of $SL_n(\mathbb C)$ as a product of $O_n(\mathbb C)\times Sym_n(\mathbb C)$ ? I mean is there a result as the polar decomposition but with orthogonal (not unitary)? thanks ...
-1
votes
1answer
46 views

Integration of hyperbolic functions. [on hold]

Kindly solve this integral. I shall be very grateful. $$ \int_0^{\infty}\frac{\mathrm{e}^{-x}}{\mathrm{e}^{ax}-1}dx $$ Thanks.
1
vote
1answer
46 views

Computing $\int_{\partial S} \frac{1}{1+z^n} dz$

Let $S=\{re^{it} : 0<r<R, 0< \varphi < 2\pi/n\}$ for some $R>1$ and $n\geq 2$. How can we compute $$\int_{\partial S} \frac{1}{1+z^n} dz?$$ I can't compute it directly, so I assume I ...
0
votes
0answers
24 views

Beyond Schwarz Lemma

Let $f(z)=a_1z+a_2z^2+a_3z^3... $ be a Schwarz function then by lemma $|a_1|\leq 1 $.But what is known about the higher coefficients? For example ; what can be said about $ max [a_1+a_2]$ ? Is there ...
0
votes
1answer
17 views

Establishing a Variant of the Mean Value Property of Harmonic Functions

Let $u:U\to \mathbb{C}$ be harmonic and $\overline{D}(P,r)\subset U$. Verify the following variant of the mean value property of harmonic functions: $$u(P)=\frac{1}{2\pi r}\int_{\partial ...
3
votes
3answers
37 views

Radius of convergence of power series of complex $\log$

Let $f(z) = \log(z)$ for $z\in \Bbb{C}\setminus (-\infty,0]$. Since $f$ is holomorphic on its domain, we know it has a power series development about each point $z_0\in \Bbb{C}\setminus (-\infty,0]$. ...
0
votes
0answers
31 views

Computing an exponential generating function from the first few terms

The current question is related to this one, and this other one. I have a number sequence, and I want to find generating ...
2
votes
1answer
33 views

Non-constant entire function-bounded or not? [duplicate]

Show that if $f$ is a non-constant entire function,it cannot satisfy the condition: $$f(z)=f(z+1)=f(z+i)$$ My line of argument so far is based on Liouville's theorem that states that every bounded ...
3
votes
1answer
45 views

A consequence of Schwarz lemma

Suppose that for some $\epsilon>0$ the function $f$ is holomorphic on $B(0,1+\epsilon)$ such that $f(a) = 0$ and $|f(z)|\leq1$ if $|z| \leq 1$. Prove for $|z| \leq 1$: $$|f(z)|\leq ...
2
votes
0answers
22 views

$f_n\to f$ iff for each closed rectifiable curve $f_n (z) \to f (z)$ uniformly for $z$ in the trace of the curve

I'd like to know if the following exercise is correct. I'm not completely sure about the last point but also I don't know what more I'd say. I really appreciate corrections or any suggestion you can ...
0
votes
1answer
18 views

If $f$ is holomorphic and $f(a) \neq 0$, then $\exists B(a,r)$ such that $f(z) \neq 0$ $\forall z \in B$

Let $G$ be a region and $f$ holomorphic in $G$. If there exists an $a$ such that $f(a) \neq 0$, then because $f$ is holomorphic, it is continuous, so there exists a $B(a,r) \subseteq G$ such that ...
0
votes
1answer
18 views

Don't understand proof of minimum modulus principle

Minimum modulus principle: If $f$ is a non-constant holomorphic function a bounded region $G$ and continuous on $\bar{G}$, then either $f$ has a zero in $G$ or $|f|$ assumes its minimum value on ...
1
vote
0answers
19 views

Irreducibility of analytic varieties

Let $V$ be an analytic variety and $V^{*}$ denote the locus of its smooth points. From Griffiths & Harris, page 21, we have that an analytic variety $V$ is irreducible iff $V^{*}$ is connected. ...
1
vote
0answers
21 views

Hölder continuity of measure associated to Nevanlinna function

Let $F$ be a Nevanlinna function and $\mu$ the (via Stieltjes inversion formula) associated measure, which is a finite Borel measure on $\mathbb R$ and let $C(\lambda)$ be the function ($\alpha \in ...
1
vote
0answers
17 views

Singular locus of analytic subvarieties

In Griffiths and Harris page 21, it is proven that the singular locus, denoted $V_{s}$ is contained in an analytic subvariety of the complex manifold $M$ not equal to $V$ which is the analytic ...
3
votes
1answer
16 views

Index of essential singularity

Let $f$ be a holomorphic function on a punctured disk $\Delta^*$ with essential singularity at puncture. Furthermore suppose that it has no zeroes on $\Delta^*$. Question: Does this integral have to ...
0
votes
1answer
21 views

Application of Residue theorem

Let f(z,w) be holomorphic in $\mathbb{C}^{n}$ and not identically zero on the w-axis. Let {$b_{j}$} be the set of singularities of f(z,w) in some disk of radius $|w| < r$. Why does the residue ...
0
votes
1answer
26 views

Schwarz-Pick type inequality for $f:\mathbb{D}\to D(0,R)$ holomorphic

Let $f:\mathbb{D}\to D(0,R)$ be a holomorphic function and let $a_i\in \mathbb{D}, 1\leq i\leq n$ such that $f(a_i)=0$ for every $i$. Show that $$|f(z)|\leq R\prod_{i=1}^n ...
1
vote
1answer
23 views

Analyticity of log f(z)

In a solution to a problem, I read that, if $f(z)$ is entire, $f(z)\neq0$ and the domain of definition of $f(z)$ is simply connected, then it is possible to choose a branch of log $f(z)$ that is ...
1
vote
4answers
24 views

Constructing a Fractional Linear Map

I am working on a practice prelim question: "Construct a nonlinear fractional map $\phi(z) = \frac{az+b}{cz+d}$ with $c \ne 0$ such that $\phi(\phi(\phi(z))) = z$. I feel like I just need to take ...
0
votes
1answer
17 views

How is the integral of $\frac{f(\zeta)-f(z)}{\zeta - z}$ over $C_{\epsilon}$ $0$?

I am trying to understand a proof of this theorem: Suppose $f$ is holomorphic in open set that contains the closure of a disk D. If C denotes the boundary circle of this disk with positive ...
1
vote
3answers
43 views

Can I use the Dirichlet's test to prove the convergence of $\sum_{n=1}^N \frac{e^{in}}{n}$?

I am trying to state that $$\sum_{n=1}^\infty \frac{e^{in}}{n}$$ converges. Is it correct that $|\sum_{n=1}^N e^{in}|\leq M$ for every positive integer $N$? I.e use $e^{in}$ as the $b_n$ term in ...
1
vote
1answer
20 views

Why can a function in $L^1(\partial \mathbb{D})$ be represented by a Fourier series?

I am looking for a reference to the claim that for any $f\in L^1(\partial \mathbb{D})$, where $\partial \mathbb{D}$ is the unit circle in $\mathbb{C}$, ...
1
vote
1answer
25 views

Function poles and divergence of series

Yesterday I tried to calculate the residues of a function the way below, but soon I realized it won't work. Now I have a question about the poles of a function, and a series representing it. $$z\in ...
0
votes
1answer
30 views

Show that $\sum_{n=1}^\infty \dfrac{1}{4^n}\sin (nz)$ comveges pointwise

Condiser $$\sum_{n=1}^\infty \dfrac{1}{4^n}\sin (nz)$$ in the region $|z|\leq1$. Show that $f_n$ converges uniformly and decide whether it converges uniformly. Well, I think that I should start with ...
0
votes
0answers
17 views

Estimate for measure associated to Nevanlinna function

Let $F$ be a Nevanlinna function (https://en.wikipedia.org/wiki/Nevanlinna_function) and let $\mu$ be the measure associated to $F$ via the Stieltjes inversion formula: ...
1
vote
2answers
25 views

Show that $\sum_{n=0}^\infty\frac{3 n+6 i}{(1+2 i)^n}$ converges.

Show that $\sum_{n=0}^\infty\frac{3 n+6 i}{(1+2 i)^n}$ converges. So I am applying the root test: $$\lim_{n\to\infty} |\sqrt[n]{\frac{3 n+6 i}{(1+2 i)^n}}|$$ and I have some difficulties with this ...
3
votes
1answer
34 views

Conformal maps onto open right half plane

On the Big Rudin there is the conformal map $$\varphi(z) = \frac {1+z}{1-z}$$ which sends $\{-1, 0, 1\}$ to $\{0, 1, \infty\}$. The book says: The segment $(-1, 1)$ maps onto the positive real ...
0
votes
0answers
27 views

Cauchy's Integral Formula Question- Calculating an integral with z^4 + 16 on the denominator

I think the first part of this question is okay. For the second part, I have found the roots and then calculated the absolute difference between these roots and i and, as they are all greater than ...
0
votes
1answer
23 views

Maximum /Minimum Modulus theorem for Harmonic Function ( Corollary 6.16 )

Suppose thatt $u(x,y)$ is a real valued non constant harmonic function on a bounded domain D. Then $u(x,y)$ can not attain its maximum or minimum value in $D$. I am studing complex ...