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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11
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2answers
132 views

Evaluating $\int_{0}^{\pi/3}\ln^2 \left ( \sin x \right )\,dx$

Good evening! I want to compute the integral $\displaystyle \int_{0}^{\pi/3}\ln^2 \left ( \sin x \right )\,dx$. However I find it extremely difficult. What I've tried is rewritting it as: ...
1
vote
1answer
15 views

Dissecting a chain of inequalities?

Refer to the following, where $R$ is some real number: I'm having a bunch of trouble following each of these steps... Can someone walk through it? Thanks.
1
vote
1answer
17 views

Parametrizing a side of a sector of a circle with natural parameterizations?

In the picture above, I'm confused as to how they parametrized $$\int_{\gamma_3}e^{iz^2}dz$$ into the part highlighted in yellow. I get that in the integral over $\gamma_2$, they simply used $z = ...
3
votes
1answer
27 views

Every compact set $S\in \mathbb{C}$ is bounded

This is my proof for every compact set $S \subseteq \mathbb{C}$ is bounded. Let $S \subseteq \mathbb{C}$ be compact and assume that it is not bounded. Then for each $z\in \mathbb{C}$ and for each ...
0
votes
0answers
17 views

when does bijective map exist for any pair of rational function?

Let me ask kind of different questions than former ones. Given $$\frac{P_1(x_1,x_2,\dots,x_n)}{P_2(x_1,x_2,\dots,x_n)}\text{, and }\frac{P_3(y_1,y_2,\dots,y_n)}{P_4(y_1,y_2,\dots,y_n)}$$ where $P_i$ ...
0
votes
1answer
31 views

complex measurable functions

I am trying to prove something about complex measurable functions. I have an idea for one direction and hope someone can give me a hint, I have gotten somee work done in this direction but need help ...
1
vote
1answer
66 views

Is $\mathbb C^3\cong \mathbb ℝ^6$?

Question: If $\mathbb C$ naturally has a bijection with $\mathbb R^2$ (and has the same cardinality as $\mathbb R$), can I just assume: That $\mathbb C^3$ has a bijection with an $\mathbb R^6$ and ...
0
votes
0answers
26 views

proof of derivative of a complex function

suppose $u(x,y)$ is harmonic in a domain $D$ and $v(x,y)$ is an harmonic conjugate of $u$. Let $f(z)=u(x,y)+iv(x,y)$. Prove $f'(z)=u_x+iv_x$.
1
vote
1answer
26 views

Why is the residue of $\dfrac{1}{z-w}R(z)$ at $w$ not continuous as a function of $w$?

Let $R(z)$ be some fixed rational function, and define $Res_R(w)$ to be the residue of $f(z)=\dfrac{1}{z-w}R(z)$ for any $w\in\mathbb{C}$. I would have thought that $Res_R$ would be continuous in ...
0
votes
1answer
38 views

$D_1(0)=\{z\in \mathbb{C} \mid |z|< 1\}$ is not compact

This is the proof I wrote for $D_1(0)=\{z\in \mathbb{C} \mid |z|< 1\}$ is not compact. $$ \bigcup_{n=2}^{\infty}D_{1-(1/n)}(0) $$ is clearly a open covering of $D_1(0)$. Consider the finite ...
1
vote
1answer
40 views

Show that $4\frac{\partial}{\partial z}\frac{\partial}{\partial\bar{z}}=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}$

I am supposed to prove the following: $$4\frac{\partial}{\partial z}\frac{\partial}{\partial\bar{z}}=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\,\,\,$$ Using the definitons ...
0
votes
1answer
23 views

An integral of Wolstenholme:$\int_0^{+\infty}\frac{\sum_1^n A_k\cos{a_k x}}{x}\mathrm {d} x$ where $\sum A_k=0$ and $a_k>0$

The book by Whittaker and Watson says it's equal to $-\sum_{k=1}^n A_k \log {a_k}$, and attibutes it to Wolstenholme. I believe this readily reduces to the simpler case of evaluating $\displaystyle ...
0
votes
0answers
32 views

Function $f(z)=\frac{\sin \pi(z-\lambda_n)}{\pi(z-\lambda_n)}$ and infinite product.

What is the relationship between the infinite product $$\prod_n \left(1-\left(\frac{z}{\lambda_n}\right)^2\right), \ \ \ \ \ z\in \mathbb C, \lambda_n\in \mathbb R$$ and the function $$f(z)=\frac{\sin ...
0
votes
1answer
30 views

The unit circle(disk), $\sigma(X)$ measurable function

If I have two measurable functions $X,Y:S \to \mathbb{R}$ (with the Lebesgue mesure) such that $X(\{x,y\})=x$ and $Y(\{x,y\})=y$ on the unit circle that is $x^2+y^2=1$. Then is $Y$ ...
1
vote
1answer
26 views

A question on the 2-norm defined by $||x||_2=\sqrt{\sum\limits_{i=1}^n|x_i|^2}$

A question on the 2-norm defined by $||x||_2=\sqrt{\sum\limits_{i=1}^n|x_i|^2}$ I am trying to prove the triangle inequality of this norm. So far I have that: \begin{align} ...
2
votes
1answer
42 views

Integrating Fresnel Integrals with Cauchy Theorem?

In regards to the above proof, I'm a little confused as to how the last conclusion was made -- How does the fact that $$\int_{-\infty}^{\infty}e^{-x^2}dx = \sqrt{\pi}$$ to conclude that: ...
1
vote
0answers
49 views

Proving Ptolemy Theorem using complex number

I am working on an assignment proving Ptolemy Theorem using complex number, and I am looking at a textbook Complex Numbers and Geometry by Hahn. Here is what I am working at this moment: THE ...
-5
votes
1answer
50 views

How to do this homework please? [closed]

This is a homework. Let $w=e^{2i\pi/7}$, $u=w+w^2+w^4$, and $v=w^3+w^5+w^6$. 1) Calculate u+v and then write $u^2$ in function of $u$. 2) Prove that $Im(u)>0$. 3) Calculate the sum: ...
3
votes
2answers
51 views

Using Cauchy's Integral Theorem to evaluate integral?

I'm going through Stein's Complex Analysis, and I'm a bit confused at one of the classical examples of using Cauchy's theorem to evaluate an integral. The example is: ...
2
votes
0answers
14 views

First order partial differential equations in complex domain

Try to solve a first order linear partial differential equation $P(x,\partial)u(x)=f(x)$ in complex domain, while the operator is of the following form: $$ ...
0
votes
1answer
26 views

Prove that $|z||b-ad| \leq M $

I need to prove the following statement: $$ |z||\frac{az + b}{z+d}-a| <= M $$ with $a,b,c,z \in \mathbb{C}, |z| \geq 1 + |d|$ and $M\geq 0$. I have reduced this to $$ |z||b-ad| \leq M $$ Also $ad ...
1
vote
3answers
29 views

Proof for inequality with complex numbers

If $a,b,c$ and $d \in \mathbb{R}$ show that $$ac+bd \leq \sqrt{a^2+b^2}\sqrt{c^2 +d^2}$$ Let's use $z=a+bi$ and $g=c+di$ so $|z|=\sqrt{a^2+b^2}$ and $|g|=\sqrt{c^2 +d^2}$. So the equation is ...
1
vote
2answers
47 views

Sufficient condition for $f(z)$ to be polynomial

I think it suffices to exhibit a sequence $\{R_n\}$ of positive real numbers such that $R_n \to \infty$ with $f(z) \neq 0,$ whenever $|z|=R_n$ and $\begin{align} ...
0
votes
1answer
24 views

Proof linear independency lemma

If $\mathbf{u}$ and $\mathbf{v}$ is in the complex vector space $V$ and $\mathbf{w}_1 = \mathbf{u} + i \mathbf{v}$ and $\mathbf{w}_2 = \mathbf{u} - i \mathbf{v}$ are linear independent then will the ...
0
votes
0answers
17 views

Show that $\delta =1$

I have a problem: Let $M \subset \Bbb C^2$ be a real analytic hypersurface: $$M=\left \{(z,w) \in \Bbb C^2 \colon \text{Im}\ w=|zw|^2+|z|^8+\frac{15}{7}|z|^2\text{Re}\ z^6 \right \}. \tag 1$$ ...
1
vote
1answer
23 views

The annulus $A_{r,s}(z_0)=\{z\in \mathbb{C} \mid r<|z_0-z|< s\}$ is open

I want to prove that the set $A_{r,s}(z_0)=\{z\in \mathbb{C} \mid r<|z_0-z|< s\}$ is open. This is my attempt. Let $z \in A_{r,s}(z_0)$. Then $|z-z_0|-r>0$. Let $r'=[|z-z_0|-r]/2$. Then ...
0
votes
1answer
17 views

Is any homeomorphism from Riemann sphere to Riemann sphere Mobius transformation?

Let $\hat{\mathbb{C}}$ be the Riemann sphere. Let $f:\hat{\mathbb{C}}\rightarrow \hat{\mathbb{C}}$ be a homeomorphism. Then, is $f$ a Mobius transformation?
0
votes
1answer
32 views

Calculating running time for C code

The problem is this: How many array accesses does the following code fragment make as a function of $N$? ...
2
votes
1answer
24 views

Quadratic formula with complex coefficients

Let $a,b$ and $c$ be complex numbers. I'm trying to prove that this version of the usual quadratic formula: $$z=\frac{-b+(b^2-4ac)^{\frac{1}{2}}} {2a}$$ solves the quadratic equation ...
7
votes
2answers
97 views

Why is this function a really good asymptotic for $\exp(x)\sqrt{x}$

$$f(x)=\sum_{n=0}^{\infty} a_n x^n\;\;\;\;\; a_n = \frac{1}{\Gamma(n+0.5)}$$ Why is this entire function a really good asymptotic for $\exp(x)\sqrt{x}$, where for large positive numbers, ...
0
votes
1answer
48 views

An entire function with an integral bound for $f'/f$ on a sequence of circles must be a polynomial

Let $f(z)$ be entire. Suppose there exists $M >0$ and sequence $\{R_n\}$ of positive real number tending to $\infty$ such that $f(z) \neq 0$ and $|z|=R_n,$ such that $\begin{align} \int_{|z|=R_n} ...
2
votes
1answer
21 views

Calculating a harmonic conjugate

Is the following reasoning correct? Determine a harmonic conjugate to the function \begin{equation} f(x,y)=2y^{3}-6x^{2}y+4x^{2}-7xy-4y^{2}+3x+4y-4 \end{equation} We first of all check $f(x,y)$ ...
0
votes
1answer
48 views

Using complex analysis to convert $b\cos \theta +a \sin \theta$ to a single trigonometric function

Using product $(a+bi)(\cos \theta+i \sin \theta) $ show that $$b\cos \theta +a \sin \theta=\sqrt{a^2 + b^2}\sin(\theta+\arctan(b/a))$$ and using this result show by induction that $$ ...
1
vote
2answers
21 views

Show: $\max_{|z|=R} \operatorname{Re}\left(z\frac{f'(z)}{f(z)}\right) \geq N $

Let $f$ be a holomorphic function defined in a neighbourhood of $\overline{D(0,R)}$ which has no zero on $\partial D(0,R).$ Let $N$ be number of zeros of $f$ in $D(0,R).$ Show: $\max_{|z|=R} ...
0
votes
0answers
9 views

What's the formal meaning of polar-coordinate partial derivatives?

I learned a technic to convert a usual integral (that is, integrator is $x^n$) to polar-coordinate integral. To do this process formally, I think one should know surface measure and Fubini's theorem ...
0
votes
1answer
12 views

Is the principal value of Argument differentiable at every nonnegative nonzero number?

How do i show that argument is continuous at points except its branch cut? I posted a question to ask whether the principal value of Argument $Arg:\mathbb{C}\setminus \{0\}\rightarrow (-\pi,\pi]$ is ...
0
votes
0answers
27 views

injectivity of holomorphic function based on its restriction

Let $f$ be holomorphic on $\mathbb D$ . Suppose there is an annulus {$z : 0 < r < |z| < 1$} in $\mathbb D$ such that $f$ restricted to this annulus is one-to-one.Show that $f$ is one-to-one ...
0
votes
0answers
34 views

$\sin (z+w)= \sin z\cos w +\cos z \sin w \forall z,w \in \mathbb C$ using Identity Theorem

In my Complex Analysis course the following problem was given after teaching "Zeros of analytic function are isolated" and the Identity Theorem. So I was supposed to solve the problem using above ...
1
vote
1answer
20 views

How do i show that argument is continuous at points except its branch cut?

Let $Arg(z)$ be the principal value of argument of $z$, so that $Arg:\mathbb{C}\setminus\{0\}\rightarrow (-\pi,\pi]$ is a function. How do I prove that $Arg$ is continuous at $z$ for all nonnegative ...
2
votes
2answers
79 views

Discrete set of zeroes of polynomials must be finite?

Let $F:\mathbb C^n\to\mathbb C^n$ be a polynomial mapping (i.e. $n$ polynomials in $n$ variables). Suppose that $Z = \left\{z \in \mathbb C^n : F(z) = 0\right\}$ is a discrete set (all points are ...
0
votes
1answer
19 views

Is it okay to ignore all other branches except principal values?

I took only a first course of complex analysis. I have learned some multi-valued functions which consistently extend the classical real functions such as Log and Trigonometric functions. However, I ...
1
vote
1answer
20 views

Analytical region of complex function $\frac{1}{z^3+a}$

Original Problem: For complex function $f(z)$: $$\frac{1}{z^3+a}$$ with $a \in \mathbb R, a > 0$ Find its region of analyticity, as well as its derivative. My question: I have trouble finding ...
0
votes
0answers
22 views

What is the definition of Riemann surface of an algebraic function?

What does it mean by the Riemann Surface of a function $y=\sqrt{x^3}$? I saw how to use the cut and glue method to obtain a sphere where $y=\sqrt{x}$ can be defined. But I was not clear in what sense ...
1
vote
1answer
71 views

Why does this function decrease at this speed?

As a part of a problem I am using this: I know that $\sum_{k=n+1}^\infty a_kz^k$ converges absolutely in a region around zero, offcourse its value at zero is zero. I also know that it is is ...
0
votes
1answer
32 views

Probable Application of Rouche's Theorem

Gamelin Exercise Problem: (Ch VIII.2): Let $f$ & $g$ be analytic functions in an open set containing a circle $C$ & its interior. Suppose $|f(z) + g(z)| \lt |f(z)| + |g(z)| ; \forall z \in ...
2
votes
0answers
52 views

If the sum of absolute values of complex numbers is at least $1$, then some subset of these numbers has absolute value at least $C$

There is a challenging problem in a book of mine on complex analysis, and I seriously do not even know where to start. I'm more than sure I don't properly understand the problem. Prove that there ...
3
votes
3answers
129 views

Find solution of equation $(z+1)^5=z^5$

I attempt to solve the equation $(z+1)^5=z^5$. My first approach is to expand the left hand side but ı get more complicated equation. So I couldn't go further. Secondly, I write equation as, since ...
1
vote
1answer
25 views

Laurent series and Bessel functions

I'm working on a problem in complex analysis that I don't know how to approach. The problem is as follows: Let the Bessel functions $J_n$, for integer $n$, be defined by ...
0
votes
1answer
32 views

On the Lebesgue measure of the set of small values of an analytic function on C

Let $f(z)$ be an analytic function on $\mathbb{C}$, $f$ is not identically zero. For each $\varepsilon>0$, I denotes ${U_\varepsilon } = \left\{ {z \in \mathbb{C}:\left| {f\left( z \right)} \right| ...
3
votes
1answer
56 views

Show that $\max(\mathrm{Re} (\exp(it)\cdot z) = |z| $

I need to show that $\max(\mathrm{Re} (\exp(it)z) = |z| $, with $t\in \mathbb{R}$ and $z\in \mathbb{C}$. Therefore I have calculated $\exp(it) = \cos(t) + i \sin(t)$. If we write $z= a+bi$, then $$ ...