0
votes
1answer
28 views

Computing ${\mathrm{d} \over \mathrm{d}t}\left(e^{it}\right)$

Let $t \in \mathbb{R}$. Is the following elementary calculation correct? $$ {\mathrm{d} \over \mathrm{d}t}\left(e^{it}\right) = \underbrace{{\mathrm{d} \over \mathrm{d}t}\left(it\right) \cdot ...
1
vote
1answer
36 views

Computing $\int_{|z|=1} {e^z \over z}\ dz$

Goal: Let $\gamma$ be the unit circle. Then I aim to compute $$ \int_{|z|=1} {e^z \over z}\ dz = \int_{\gamma} {e^z \over z}\ dz $$ Attempt: Consider that $\gamma$ is a closed curve. Let $a = 0$. ...
1
vote
2answers
26 views

Showing that ${d \over dz}\log\left[ z - a \over z - b \right] = {1 \over (z - a)} - {1 \over (z - b)}$

I'm trying to show that $$ {d \over dz}\log\left[ z - a \over z - b \right] = {1 \over (z - a)} - {1 \over (z - b)} $$ However my attempt yields that $$ {d \over dz}\log\left[ z - a \over z - b ...
1
vote
0answers
42 views

Cauchy Integral Theorem problem (lack of understanding)

First of all i was asked to evaluate this integral $\int_\gamma \frac{2z}{(z-1)(z-3)} dz$ where $\gamma (t) = 2e^{it}$ for $0\leq t \leq 2\pi$. Now I thought I would have to calculate this ...
2
votes
1answer
47 views

Integral $\int_0^{\pi/2} \log^n (\sin t)\log^p (\cos t) dt$

I am looking for a closed form expression for the logarithmic trigonometric integral $$ I_{n,p}=\int_0^{\pi/2} \log^n (\sin t)\log^p (\cos t) dt \quad (n\geq 0, p\geq 0). $$ Closed form expression ...
5
votes
2answers
95 views

Integrate $I=\int_0^1\frac{\ln x}{x^n-1}dx$

Hi I am trying to obtain a closed form for$$ I_n=\int_0^1\frac{\ln x}{x^n-1}dx, \quad n\geq 1. $$ This integral is quite nice and generates many other known closed form results such as $$ ...
0
votes
0answers
18 views

integrating of complex exponential function

I know $\int x^a dx=\frac{x^{a+1}}{a+1}$ when $a$ is real. How I can calculate this integral when $a$ is complex?
0
votes
3answers
49 views

Algebraic Equation?

$$Ve^{i\theta} = We^{i\phi}$$ where, $V$ and $W$ are some real constants. From this my book concludes: $\theta = \phi$. How does it conclude this? I don't see why its valid to just equate the ...
2
votes
1answer
28 views

functions of two variables with one variable defined on a compact set uniformly converge to zero

Let $f$ be a holomorphic function on $[0,1]\times \mathbb{R}$. If for each $x\in [0,1]$ fixed, $\lim_{y\to\infty}f(x,y)=0$, prove that $f$ is bounded. My idea: I do not know how to prove and I also ...
0
votes
2answers
43 views

Multiplying and Dividing Series

For example, how do you compute the taylor series for $$e^x \sin x=\sum_{n=0}^{\infty} \frac {x^n}{n!} \sum_{n=0}^{\infty} (-1)^n\frac {x^{2n+1}}{(2n+1)!}$$ Of course I want the result to contain ...
0
votes
1answer
69 views

Why isn't $i$ affected by powers?

When finding roots of complex functions we can write for example: $$z=2-2i$$ Let's find complex numbers $w$ such that $$w^4 = 2-2i$$ $$\large z = \sqrt{8} e^{ \frac{- \pi }{4} i}$$ This reads: ...
0
votes
0answers
14 views

conflictions of analytic functions to the boundary and Schwarz reflection principle

Let $\Omega$ be an open subset of $\mathbb{C}$ and $f:\Omega\longrightarrow \mathbb{C}$ be a holomorphic function. Then for any $z\in \Omega$ and any $r>0$ such that $D(z,r)\subseteq \Omega$, $f$ ...
1
vote
0answers
17 views

Cauchy integrals over a line

Can we generalize the Cauchy integral formula from a circle to a line? Since for real integrals, the following types of improper integrals do not converge, is it correct or not that for $z\notin ...
2
votes
1answer
24 views

Examples of vector field that is continuously differentiable but not conservative?

I am just curious what would be the case in which a vector field ($\vec f :\Bbb R^2 \rightarrow \Bbb R^2$) is well-defined and continuously differentiable on a region R enclosed by a simple closed ...
4
votes
0answers
75 views

Ugly-nice double series

I'm trying to evaluate the following ugly double sum (presented in raw notation as used in my calculations): $\sum _{m=1}^{\infty } \sum _{n=1}^{\infty } \frac{4 m \cos \left(\frac{2 \pi m ...
0
votes
0answers
19 views

How to find the tangential component of Velocity?

Let $v=-\nabla \phi$, where $\phi$ is the velocity potential. I am interested in to find the value of $|v_{tan}|$ which is the tangential component of velocity.
1
vote
1answer
83 views

About asymptotic behaviour of a divergent integral.

I have the function $f(x) = x \tanh(\pi x) \log (x^2 +a^2)$ where $a$ is some positive real number. For the logarithm I am assuming a branch-cut along the positive imaginary axis starting at $x = ia$. ...
1
vote
1answer
49 views

consequences of Schwarz lemma of holomorphic functions of unit disk

Let $D$ be the open unit disk centered at $0$ in the complex plane. Let $f:D\longrightarrow D$ be holomorphic such that $f(0)=0$. Use the Schwarz lemma to prove that $|f(z)+f(-z)|\leq 2|z|^2$ for any ...
0
votes
2answers
48 views

conformal map/Mobius transformation from annulus to $\mathbb{C}\setminus \overline{D(0,1)}$

Does there exist a conformal bijection/Mobius transformation from the open unit disk to the whole complex plane? Does there exist a conformal bijection/Mobius transformation from the annulus $\{z\in ...
0
votes
1answer
52 views

proving a limit of a function by definition

Consider $f: \Bbb{C} \to \Bbb{C}$ defined by $$ f(z) = \begin{cases} z^3 + 2z &\text{if } z \ne i \\ 3 + 2i &\text{if } z = i \end{cases} $$ Prove that $$ \lim_{z \to i} f(z) = i $$ using the ...
0
votes
1answer
41 views

Complex Fourier series of $f(\theta) = e^{\theta}$

I have the following Fourier series problem: Let $f(\theta)$ be the periodic function such that $f(\theta) = e^\theta$ for $-\pi<\theta\leq\pi\;$, and let ...
0
votes
0answers
30 views

Openness of Sets - Point set Topology

What's the formal method called where we find a $p>0$ such that $|(x,y)-(x_0,y_0)|$ is less than $p$?
0
votes
1answer
29 views

Complex polynomial decomposition - Residue Theory

I am given the following function: $R(z) = (z^2-9)/(z^2+9)^2 $ I need to let $R = P/Q$ be a rational function with $deg P < deg Q$. I will let $ξ$ be a pole of $R$ and the coefficient of $1/(z-ξ)$ ...
0
votes
0answers
25 views

If $Z$ is an admissible function, does $f(z) = f(|z|)$?

If $Z$ is an admissible function, does $f(z) = f(|z|)$? For example, if $f(z) = x^3 + 1$ and I am given $2$ points $z_1 = -1+i\sqrt3$ and $z_2 = -1-i\sqrt3$, can I just find the moduli and use that ...
1
vote
1answer
66 views

Disproving the mean-value theorem of calculus to complex functions?

I'm defining a function $f(z) = z^3 + 1$, and I will let $2$ points $z_1 = (-1+i\sqrt3)/2$ and $z_2 = (-1-i\sqrt3)/2$ I am trying to show that there is no point $w$ on the line segment from $z_1$ to ...
3
votes
2answers
72 views

erf(a+ib) error function separate into real and imaginary part

Is there an easy way to separate erf(a+ib) into real and imaginary part?
2
votes
2answers
56 views

Find all singularities of a function

Let $f(z)=z(e^{\frac 1 z} -1)\tan{\frac 1 {z-1}}$. Find all zeros and singularities of $f$. I know that $f$ is analytic in $\{z:z\not=0,1,\frac 1 {\pi k+\frac \pi 2}\}$ where $k\in\mathbb{Z}$ and that ...
1
vote
1answer
39 views

The definition of Residue

In Wikipedia the definition of a residue of a function $f$ in a point $a$ is a unique value $R$ such that $f(z)-\frac{R}{z-a}$ has an anti derivative in a punctured disk $0<|z-a|<\delta$. How is ...
0
votes
1answer
32 views

Analytical Formula for Hilbert Transform of a Ricker Wavelet

I am attempting to validate some numerical code I have to compute Hilbert transforms. As I am interested in the Hilbert transforms of functions with rapid decay, I wanted to unit test my code with the ...
0
votes
1answer
37 views

Why does $\sum_{n\neq0}\:\left|\frac{a_n}{n}\right| \leq \sqrt{\sum_{n\neq0}\frac{1}{n^2}}\sqrt{\sum_{n\neq0}|a_n|^2}$?

My question is: Why does $$\sum_{n\neq0}\:\left|\frac{a_n}{n}\right| \leq \sqrt{\sum_{n\neq0}\frac{1}{n^2}}\sqrt{\sum_{n\neq0}|a_n|^2},$$ where $a_n$ is some complex number, $n$ an integer going ...
0
votes
0answers
22 views

Providing an upper bound for a sum of complex numbers

Let $(\alpha_l)_{l=0}^{k-1}$ and $(\beta_l)_{l=0}^{m-1}$ be two sequences of complex numbers where $m>k$. It is known that $$ 0<\frac{1}{m}\sum_{l=0}^{m-1}|\beta_l|^2\leq A $$ where $A>0$. ...
1
vote
2answers
63 views

Contour Integral for Cosine and a rational function

I've been trying to figure out this integral via use of residues: $$\int_{-\infty}^{\infty} \displaystyle \frac{\cos{5x}}{x^4+1}dx$$ The usual semicircle contour wont work for this guy as the ...
3
votes
2answers
58 views

Differentiate complex function?

$$f(z)=3z^2+\bar{z}$$ I want to show the function is either differentiable or not so I can state if it is holomorphic or not. What is the method for this ? Edit - Can some give an example of how to ...
0
votes
3answers
35 views

About the Scalar product

These are the lecture notes of my teacher and I am getting confused how he reached at $V_1$.$V_2$= Re($z_1$$z_2$). Can anyone help me to understand this.
3
votes
2answers
76 views

Elimination of Trigonometric Functions

Is there a simple way to eliminate the trigonometric functions here? $$ \begin{array}{lcl} A\cos(3\omega\tau)+B\sin(3\omega\tau)+C\cos(\omega\tau) &=& D\\ ...
3
votes
2answers
109 views

Trouble with $\int_0^\infty e^{-ix^2}\mathrm{d}x$

I'm trying to evaluate $$ \int_0^\infty \mathrm{d}x\ e^{-ix^2}. $$ I tried to integrate on the following contour $\Gamma_R$: the frontier of a circular sector, centered at the origin, of angle $\pi / ...
3
votes
2answers
64 views

Justifying an ODE's solution

In an introductory lesson into ODEs, in order to "semi-rigorously" justify the solution for e.g. : $(a)\ \ y'+y=0$ we proceed without an ansatz or guess solution (hence the "semi-rigour"): Let: ...
3
votes
2answers
76 views

Asymptotics of coefficients

This is a question that asks the reader for a $strategy$ to solve a particular problem. I cannot solve this problem myself so I am looking around for general methods one might use to confront it with. ...
1
vote
1answer
25 views

Show that the composition of the two functions is the identity.

I have to check that the composition of the following functions gives the identity (or that one function is the inverse of the other): $$\pi:S^2\backslash \{N\}\to\mathbb{C}$$ $$(x_1,x_2,x_3) ...
0
votes
2answers
89 views

Prove that $f(z) = \sum\limits_{k = 1}^\infty \frac{z^{2^k}}{2^k}$ is continuous in the closed unit disc and holomorphic inside it.

I have started off by assuming that there is a disc of radius $r$ for which $|z|<r$ for $r \in (0,1)$ and $z \in D_r$. This implies that $|z|^{2^k} < r^{2^k}$ And after that, I don't know ...
3
votes
1answer
131 views

Laplace transform of and impulse sampled function using “frequency” convolution

This is a long question, but assume we have this: The book uses the frequency convolution theorem to solve this problem. To solve the integral, it uses a contour + residue theorem to solve it. The ...
2
votes
1answer
38 views

Continuous functions. Second norm

Let $f:[a,b]\rightarrow \mathbb{R}^{d}$ be continuous. I need to prove that $\left \| \int_{a}^{b}f(x)dx \right \|_{2}\leq \int_{a}^{b}\left \| f(x)) \right \|_{2}dx$.
3
votes
1answer
30 views

number of zeros in a disk of a holomorphic function

Let $f$ be a holomorphic function defined in a beighborhood of $\overline{D(0,R)}$ which has no zeros on $\partial D(0,R)$. Let $N$ be the number of zeros of $f$ inside $D(0,R)$. Prove that ...
3
votes
1answer
31 views

Superior limit of integrals of entire functions

Let $f$ be an entire function on $\mathbb{C}$. If $f$ is not constant, then I want to prove \begin{equation} \limsup_{R\to\infty}\int_{\lvert z\rvert=R}\lvert f(z)\rvert\,\lvert dz\rvert=\infty. ...
0
votes
1answer
30 views

How to find the value of tangent vectors?

In the figure $z_1=a_1+ib_1$ and $z_2=a_2+ib_2$ are the two curves. $T_1$ and $T_2$ be the tangents on the curves $z_1$ and $z_2$. What I am interested to know what will be the tangent vectors?
0
votes
2answers
47 views

logarithm of an entire function

Let $f$ be an entire function (holomorphic over the complex plane). If $f$ has no zero point, then $\text{Log} f$ is also an entire function. How to prove this? My idea: one branch of $\text{Log}f$ ...
1
vote
0answers
70 views

Integrate: $\int\limits_0^\infty{\frac{x^{n-2}}{b\left(1+ ~a x^{\frac{n-1}{n-2}}\right)} \sin{(x b)}~ dx}$

I am trying to solve the integral: $\int\limits_0^\infty{\frac{x^{n-2}}{b\left(1+ ~a x^{\frac{n-1}{n-2}}\right)} \sin{(x b)}~ dx}$ where $x$ is real and $a, b, n$ are positive real constants. I ...
7
votes
2answers
302 views

Is there an analytic function satisfying $\,\,f\big(\!\frac 1 n\!\big)=\frac 1 {\sqrt{n}},\, \,n\in\mathbb N$?

Is there a function that is analytic in an open neighbourhood of $z=0$ and satisfies $$f\left(\!\dfrac 1 n\!\right)=\dfrac 1 {\sqrt{n}},$$ for all natural number $n$? I guess this problem requires ...
1
vote
0answers
37 views

entire function with no zeros. Then $f$ must be of the form $f(z)=\exp(g(z))$ [duplicate]

Let $f$ be an entire function with no zeros. Then $f$ must be of the form $f(z)=\exp(g(z))$ where $g$ entire. Is it true or not? How to prove?
0
votes
1answer
67 views

some characterizations of entire functions to be a polynomial

Let $f$ be an entire function. Then under which conditions will $f$ be a polynomial? How to prove under either of the following conditions, $f$ is polynomial? (1). $|f(z)|>1$ whenever $|z|>1$; ...