0
votes
0answers
14 views

“Angle-preserving” equivalent to conformal?

I'd like to investigate the common turn of phrase that conflates "angle-preserving map" with "conformal map". Let $f:\Bbb R^2\to\Bbb R^2$ be a continuous function. I'll define $f$ to be ...
2
votes
3answers
50 views

Conplex/real Integration and poles of function

So I am working on the following problem: Let $\Delta $ be the unit disk centered at origin, and $f$ is holomorphic on $\Delta-\{0\}$. If $$\int_\Delta|f|dxdy<\infty$$ show that $f$ has at most a ...
0
votes
1answer
43 views

(absolute) Convergence of a series

I want to prove that the following series is convergent for $x>0$: $$ \sum_{n=1}^\infty \left( \prod_{p\mid n} \frac{1}{p-1}\right) n^{-x} $$ I tried to estimate the product but I didn't get so ...
0
votes
0answers
23 views

asymptotics from Laplace transform

Suppose I know that a non-negative random variable with density $f$ has the following Laplace transform: $$\hat{f}(s)=\int_0^{\infty}e^{-st}f(t)dt=\frac{1}{\cosh(\sqrt{2s}x)}$$ where $s>0$ and ...
0
votes
1answer
27 views

What does monotonically convergent mean in this example.

Suppose that $$f(x)=\sum_{n=1}^{\infty}f_n(x)\,\,\,\,\,\,\,\,\,\,(x\in X)$$Where $f_n:X\rightarrow [0,\infty]$ for $n=1,2,3...$ Let $g_N=f_1+...+f_N$. Then the sequence $\{g_N\}$ converges ...
1
vote
2answers
24 views

$f_n$ converges uniformly on $\overline\Omega$

Suppose $\Omega$ be a bounded region and $\{f_n\}_{n\in\mathbb N}$ a sequence of continuous functions on $\overline\Omega$ which are holomorphic in $\Omega$ and $f_n$ converges uniformly on the ...
1
vote
3answers
40 views

Biconditionality-ish of Epsilon Delta Proof of a Limit?

I have known the precise ($\epsilon$, $\delta$) definition of a limit, $$\lim_{x\rightarrow a} f(x) = L \iff \forall \epsilon>0,\exists\delta>0 : (0<|x-a|<\delta \implies ...
0
votes
0answers
30 views

Roots of polynomial

I came across when reading paper: Given $f'(z)+\alpha zf''(z) + \gamma z ^2f'''(z) $ where $\mu = \tfrac{(\alpha-\gamma)-\sqrt{(\alpha-\gamma)^2-4\gamma}}{2}$,$\quad$ $\nu+\mu=\alpha-\gamma$, ...
0
votes
0answers
34 views

Understanding the statement that $\varphi(\emptyset)=0$ implies $\varphi$ is not identically $\infty$

The proposition is from "Real and Complex Analysis" by Rudin.It states: Let $s$ be a nonnegative measurable simple function on $X$ . For $E\in\mathfrak M$ (where $\mathfrak M$ is a $\sigma$-algebra ...
0
votes
0answers
21 views

Topological conjugacy in Hénon map

$\textbf{Definition:}$ $\textit{(Topologically conjugate)}$ Let $f:A\rightarrow A$ and $g:B\rightarrow B$ be two maps. $f$ and %g% are said to be topologically conjugate if there exists a ...
0
votes
0answers
13 views

I need some clarification on the term of “measurable on” and “continuous on”

I run across theorems similar to this one: "If $f$ is a complex measurable function on $X$, there is a complex measurable function on $X$ called $\alpha$ such that $\vert\alpha\vert=1$ and ...
0
votes
1answer
22 views

Is the split normal distribution analytic on $\mathbb{C}$?

I wonder if the split normal distribution which expressed as following is analytic on $\mathbb{C}$ or not? $ p(x)= \left\{ \begin{array}{l l} \frac{2}{1+\gamma} \cdot \frac{1}{\sqrt{2 \pi}} ...
0
votes
0answers
9 views

Show $\overline{\lim} |b_n| < \infty$ when $\lim e^{it b_n}$ exists for all $|t| < \delta, \delta > 0$

$\{b_n\}$ is a sequence of real numbers. If $\lim e^{it b_n}$ exists for all $|t| < \delta, \delta > 0$, show that $$\overline{\lim} |b_n| < \infty$$ How can prove this statement? I ...
3
votes
2answers
181 views

Cauchy product $\sum_{n=-\infty}^{\infty}\sum_{k=-\infty}^{\infty}a_{n-k}b_k$

I have been told that, if $\{a_n\}_{n\in\mathbb{N}}$, $\{a_{-n}\}_{n\in\mathbb{N}^+}$, $\{b_n\}_{n\in\mathbb{N}}$ and $\{b_{-n}\}_{n\in\mathbb{N}^+}$ are absolutely summable complex sequences, ...
0
votes
1answer
32 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 ...
0
votes
0answers
30 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 $f'(z)=u_x+iv_x$.
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 ...
7
votes
0answers
101 views

Another way of expressing $\sum_{k=0}^{n} (-1)^k\frac{H_{k+1}}{n-k+1}$

In this post Another way of expressing $\sum_{k=0}^{n} \frac{H_{k+1}}{n-k+1}$ I asked for a solution of the non-alternating series. How about the alternating series? Can we find a nice way of ...
0
votes
1answer
34 views

Appolonius circle with $\infty$ as one limit point.

For two complex numbers $a,b$ circle of appollonius with limit point $a$ and $b$ are given by $$\left\lvert \frac{z-a}{z-b}\right\rvert=r.$$ I can only see these circles when limit points are finite ...
0
votes
1answer
33 views

Why is the set $E$ measurable?

I am having trouble understanding a proof presented in Rudin's Real and Complex Analysis. The theorem states, if $f$ is a complex measurable function on $X$, there is a complex measurable function ...
0
votes
1answer
37 views

Why does these complex sequences converge uniformly?

I have one complex series and one sequence. It is used in complex analysis in a part of my book where they are integrated. However, as you know in order to change limit and integration order it has to ...
2
votes
0answers
30 views

Interchange of infinite product and limit

The Problem Let $(a_{n,m})_{n,m\in \mathbb{N}}$ be an sequence of complex numbers. Under which conditions can I interchange product and limit? $\lim_{m\to\infty}\prod_{n=1}^{\infty} ...
11
votes
5answers
179 views

Another way of expressing $\sum_{k=0}^{n} \frac{H_{k+1}}{n-k+1}$

Do you know any nice way of expressing $$\sum_{k=0}^{n} \frac{H_{k+1}}{n-k+1}$$ ? Some simple manipulations involving the integrals lead to an expression that also uses the hypergeometric series. ...
1
vote
0answers
53 views

Suggested book for self study.

I have a degree in Financial Risk Management, and did 4 semesters of calculus and analysis(but that was about 10 years back), with most of my other efforts going toward Mathematical Statistics and ...
3
votes
2answers
34 views

If $a_n\to0$, there exists $\pm$ such that $\sum\limits_n\pm a_n$ converges [duplicate]

Our Analysis I lecturer in his last lecture for the course gave us a problem to think about. I've been thinking about it for a while and has been bothering me for some time. It looks like a ...
5
votes
1answer
116 views

Prove $\int_0^1 \frac{\ln(1+t^{4+\sqrt{15}})}{1+t}\mathrm dt= -\frac{\pi^2}{12}(\sqrt{15}-2)+\ln (2) \ln(\sqrt{3}+\sqrt{5})+\ln(\phi) \ln(2+\sqrt{3})$

Prove that: \begin{equation} \int_0^1 \frac{\ln\left(1+t^{4+\sqrt{15}}\right)}{1+t}\mathrm dt= -\frac{\pi^2}{12}(\sqrt{15}-2)+\ln (2) \ln(\sqrt{3}+\sqrt{5})+\ln(\phi) \ln(2+\sqrt{3}) ...
2
votes
2answers
81 views

Why is continuous differentiability required?

I have two questions. My book proves that if $f:\mathbb{C}\rightarrow \mathbb{C}$ is a holomorphic function, then it satisfies the Cauchy-Riemann equations, and if we look at the function as $F: ...
1
vote
1answer
42 views

hexic polynomial question

I am faced with a polynomial of the form $$ ax^6+bx^3+cx+d=0, $$ where the coefficients are complex. I want to be able to say something about the roots of this polynomial (including finding them!). Is ...
5
votes
0answers
109 views

Is there a book only about epsilon delta proofs?

I want to know if there is such book, with beautiful epsilon delta proofs of all kind.
2
votes
0answers
49 views

$f\in C^\omega ((a-R,a+R),\mathbb{R})$ [closed]

We discuss the following question in the field of real numbers . A a power series $f(x)= \sum_{n=0}^\infty a_n (x-a )^n$ converges in $(a-R,a+R).$ Prove: $$\forall x_0\in\left(a-R,a+R \right), ...
1
vote
0answers
49 views

Fourier transform and inverse transorm

I have to prove that for $f\in L^1(\mathbb{R})$ $$ \check{\hat{f}}=\hat{\check{f}}, $$ where $$ \hat{f}(\xi):=\int\limits_{\mathbb{R}}e^{-i\xi x}f(x)\mathrm{d}x $$ and ...
1
vote
0answers
25 views

separate vs joint real analyticity

Let $$f(x,y) := xy\exp\left(-\frac{1}{x^2+y^2}\right),$$ if $(x,y)\neq (0,0)$ and $f(0,0):=0$. I read the claim that $f$ is (a) separately real analytic on $\mathbb{R}\times\mathbb{R}$ (i.e. for ...
1
vote
1answer
49 views

If holomorphic $\{f_n\}\to f$ uniformly on compact subsets of $U$, then do $f_n$ and $f$ eventually have the same number of zeros?

Let $U$ be an open subset of $\mathbb{C}$. Let $\{f_n\}$ be a sequence of holomorphic functions on $U$ such that $f_n\to f$ uniformly on any compact subset $K$ in $U$. Suppose $f$ is not constant, ...
2
votes
2answers
56 views

For holomorphic functions, if $\{f_n\}\to f$ uniformly on compact sets, then the same is true for the derivatives.

Let $\Omega$ be an open subset in $\mathbb{C}$. Let $\{f_n\}$ be a sequence of holomorphic functions on $\Omega$ such that $f_n\to f$ pointwise and converges uniformly on any compact subset ...
5
votes
1answer
44 views

how to determine the existence of double limit?

Let $f(x,y)$ be a function of two variables. Are there any criterions to determine the existence of double limit $$ \lim_{(x,y)\to(x_0,y_0)} f(x,y)? $$ If for all $y\in(y_0-\delta,y_0+\delta)$, ...
0
votes
0answers
35 views

show that a function belongs to $L^1(m)$

Let m be normalized lebesgue measure on $d{\Bbb D}$ and for $|z|<1$ and $|w|=1$, let $p_z(w)=(1-|z|^2)/|1-\bar{z}w|^2$. I need to show that a- $p_z \in L^1(m)$ b- for every $x^*\in (L^1(m))^*$, ...
2
votes
2answers
73 views

A function is real-differentiable iff it has a complex-differentiable extension

Is this conjecture true? A function $f:\Bbb R\to\Bbb R$ is real differentiable at $a$ if and only if there exists a complex-differentiable function $g:A\to\Bbb C$ for some neighborhood of $a\in ...
0
votes
2answers
66 views

Show that $e^{iy} = 1 + iy + \frac{\mu_1 y^2}{2}$ for all $y \in \mathbb{R}$ with $|\mu_1| \leq 1$

How to show the expansions \begin{gather*} e^{iy} = 1 + iy + \frac{\mu_1 y^2}{2}\\ e^{iy} = 1 + iy - \frac{1}{2}y^2 + \frac{\mu_2 |y|^3}{3!} \end{gather*} where $y \in \mathbb{R}$ and $|\mu_1| \leq 1$ ...
1
vote
1answer
36 views

Assume that $f$ is analytic and one-to-one on $\mathbb{D} = \{z : |z| < 1\}$ and $f(z) = z + z^2g(z),$ where $g$ is analytic in $\mathbb{D}.$

Assume that $f$ is analytic and one-to-one on $\mathbb{D} = \{z : |z| < 1\}$ and $f(z) = z + z^2g(z),$ where $g$ is analytic in $\mathbb{D}.$ Prove that if $f(\mathbb{D})⊂\mathbb{D}$ or ...
1
vote
1answer
41 views

Uniformly analytic functions

Consider the following definition: Let $\Omega$ be an open set of $\mathbb{R}_x^n$, $x = (x_1, ..., x_n)$. A $\mathcal{C}^{\infty}$-function $\varphi(x)$ on $\Omega$ is said to be uniformly analytic ...
4
votes
1answer
95 views

What is $\zeta(n)$ as $n$ tends to $\infty$? How fast it goes to the limit?

What is $\zeta(n)$ as $n\to\infty$? How fast it goes to the limit?
2
votes
1answer
56 views

an inequality derived from conformal automorphisms of unit disk

Let $f$ be a holomorphic function on $D(0,1)$ such that $|f(z)|<1$ for all $z\in D(0,1)$. I have obtained $$ \frac{|f(0)|-|z|}{1+|f(0)||z|}\leq |f(z)|\leq \frac{|f(0)|+|z|}{1-|f(0)||z|}. $$ Is it ...
-1
votes
1answer
37 views

Differentiability: Partially Defined Functions

These ideas came to my mind while reading Lee's Introduction to Smooth Manifolds. (Cf. discussion on p. 45.) Also note that though I were able to resolve the first problem the second one is still ...
1
vote
1answer
34 views

The functions $\{f_n(x) = n\}$ are analytic and each miss the points $-2, -3$. But, they are not a normal family. So what am I missing. Thanks.

Here is a theorem of Montel: Let $\mathcal{F}$ be a family of analytic functions defined on a domain $\Omega$ . If there are two fixed complex numbers $a$ and $b$ that are omitted from the range of ...
1
vote
1answer
58 views

Let $f(z) = \sum_{n = 0}^\infty a_nz^n$ be the Taylor series around $0$. Prove that lim $a_n/a_{n+1} = z_0.$ [duplicate]

Let $f(z) = \sum_{n = 0}^\infty a_nz^n$ be the Taylor series around $0$ of a function which is analytic in $\mathbb{C}$ \ ${z_0}$, $z_0\neq 0$ and has only a simple pole at $z_0.$ Prove that $lim_{n ...
1
vote
0answers
62 views

To show a power series is a Taylor series

Is it possible to prove if $f(x) = \sum_{n = 0}^\infty a_n(x - a)^n$ then the series is the Taylor series of $f$ without using complex analysis, as done here?
11
votes
0answers
220 views

The closed form of $\int_0^{\pi/4}\frac{\log(1-x) \tan^2(x)}{1-x\tan^2(x)} \ dx$

What tools, ways would you propose for getting the closed form of this integral? $$\int_0^{\pi/4}\frac{\log(1-x) \tan^2(x)}{1-x\tan^2(x)} \ dx$$
2
votes
2answers
166 views

How to calculate $\int_{-\infty}^\infty\frac{x^2+2x}{x^4+x^2+1}dx$?

I want to calculate the following integral: $$I:=\displaystyle\int_{-\infty}^\infty\underbrace{\frac{x^2+2x}{x^4+x^2+1}}_{=:f(x)}dx$$ Of course, I could try to determine $\int f(x)\;dx$ in terms of ...
0
votes
2answers
38 views

solving $|(x-3)(x-1)| $$\le$ $|\frac{1-x}{x-3}|$ graphicly [closed]

how to solve $|(x-3)(x-1)| $$\le$ $|\frac{1-x}{x-3}| $ in the graphic method?
0
votes
1answer
33 views

Isoperimetric inequality with Green-capacitiy

I was wondering what the progress is, in isoperimetric inequalities for Capacities, specifically with the Green kernel ( optional: and Riesz kernel with $a\in (2,\infty)$). Or if it is solved already, ...