0
votes
1answer
31 views

the absolute value of $\frac{1}{e^{i\omega t}-1}$

I am told to get the absolute value of $$\frac{1}{e^{i\omega t}-1}$$ I sense that there's something ridiculously simple about this, but I tried working from the fact that if I square it, the absolute ...
5
votes
1answer
77 views

If $e^{i\theta}=e^{i\varphi}$, then $\theta-\varphi=2k\pi$

This is pretty easy I think but I am having a tough time trying to prove this in a satisfying way to me. I am trying to show that $$e^{i\theta}=e^{i\varphi} \Rightarrow \theta-\varphi=2k\pi,\, \text{ ...
0
votes
0answers
45 views

Complex exponential integral: Mathematica and MATLAB give unexpected results

I currently compare analytical vs. numerical evaluation of the complex exponential integral and find mismatches: The imaginary part differs by $\pm \pi$ and the real part has a large error when ...
2
votes
2answers
58 views

Evaluating exponential integral

I am struggling for some time to solve the following integral: $$ \int_{-n}^{N-n} \left( \frac{e^{-j\pi(\alpha-1)\tau}}{\tau} - \frac{e^{-j\pi(\alpha+1)\tau}}{\tau} \right) d\tau $$ $N$ is a ...
0
votes
2answers
41 views

Construction of complex numbers and exponent rules for them

I have some questions about the construction of the complex numbers in this Wikipedia article, especially of the exponents of complex numbers. $1$. Is it enough to define it as $a+bi$, where a,b are ...
2
votes
0answers
37 views

Functional Equation involving derivatives and time-steps [duplicate]

I am attempting to solve the equation $$f(x + 1) = f'(x)$$ for distributions $C \rightarrow C: f(x)$ My first guess to exploit the fact that this seems similar to identity $$\sin\left( ...
0
votes
1answer
40 views

Choosing a branch of the square root

Assume $O$ is the compliment of the non-positive part of the real line to the complex plane. This is an open and connected set. Only one of the values of $\sqrt z$ in $O$ has positive real part. With ...
2
votes
0answers
46 views

Why does the Riemann Xi function $(\xi(s))$ have order of growth 1

Why does $s(s-1)\xi(s)$, have order of growth 1? In other words, why is it that $\forall \epsilon > 0 $ $\exists A_{\epsilon},B_{\epsilon} \in \mathbb R_+$ so that $\forall s \in \mathbb C$, ...
2
votes
1answer
32 views

Finding Complex Zeros

I have to find how many zeros $3e^z - z$ has in $abs(z) < 1$. I know a function has a zero of order m if $f(z) = (z-z_0)^mg(z)$, where $g(z)$ does not equal 0. I was thinking of maybe applying ...
0
votes
2answers
45 views

Expressing the sine function in terms of exponential

Prove $e^{iz} - e^{-iz} = \sin z$. I used $$\begin{align*} \sin z & = z - z^3/3! + z^5/5! - z^7/7! + \dots & (i) \\ e^{iz} & = 1 - z^2/ 2! - iz^3/3! + \dots & (ii) \\ e^{-iz} ...
0
votes
0answers
179 views

Laurent Series and Taylor Expansion of $ 1 / (e^z - 1) $

Could someone please assist me with the second part of the second paragraph, from "By expanding $f_1$..."? I am not convinced that my expansion for $f_1$ is right - I used the standard binomial, ...
4
votes
3answers
270 views

Find a closed form from the given power series

I have the power series $\sum_{n=0}^{\infty} {z^{2n}\over{n!}}$, how do I find the closed form for this power series. I am aware that $e^z=\sum_{n=0}^{\infty} {z^{n}\over{n!}}$, so I tried to ...
0
votes
1answer
110 views

An Application of Rouche's Theorem to Two Cases

Here is my question - it is an example sheet question, completely non-examinable: [I have managed this first part, but am including it to help give a sense of where the question is going.] $(i)$ ...
0
votes
1answer
86 views

How does the complex exponential function transform the unit circle?

I know you can write every complex number on the unit circle as $e^{i\theta} = \cos(\theta)+i\sin(\theta).$ But what does it look like when you raise $e$ to the values? You get ...
0
votes
1answer
31 views

Multi part problem to prove functional relation of the exponential function

I'm worried about part (i) right now mostly. Part 3 is easy, and part 2 I can probably get after some work. I know that $\exp(-z) = \large\sum\limits_{n=0}^\infty \frac{-z^n}{n!} = ...
3
votes
1answer
163 views

How can we describe the graph of $x^x$ for negative values?

We usually only see the graph $y=x^x$ for $x>0$, because $x^x$ is a complex number for most negative values of $x$. Yet here is a full graph of $y=x^x$ on the real line: This graph may seem like ...
0
votes
2answers
57 views

Calculate the integral $\int_{\varGamma}\frac{3e^{z}}{1-e^{z}}dz$

I am looking to solve $$\int_{\varGamma}\frac{3e^{z}}{1-e^{z}}dz,$$ where $\varGamma$ is the contour $|z|=4\pi/3$. We have been asked first to consider $e^{z}=1$ and $e^{z}=-1$ which I get to be ...
1
vote
1answer
95 views

The transformations on the nome and Landen's transformation

Could someone please explain how to transform the nome $q = e^{-\pi K'/K}$ from $q^2$ to $q$ and then to $-q$? In other words, how does changing $q^2$ to $q$ and then $q$ to $-q$ affect $k$ and $K$. ...
1
vote
2answers
44 views

Need help with a proof concerning zero-free holomorphic functions.

Suppose $f(z)$ is holomorphic and zero-free in a simply connected domain, and that $\exists g(z)$ for which $f(z) =$ exp$(g(z))$. The question I am answering is the following: Let $t\neq 0$ be a ...
25
votes
3answers
815 views

If there are entire $G_k$s such that $f=\exp\circ\exp\circ\cdots \circ\exp\circ G_k$ ($k$ times), must $f$ be constant?

I am a French guest and I hope that my English isn't too bad... So here is my issue: I consider an entire function $f$ which satisfies the following property for each complex number $z\in ...
0
votes
2answers
75 views

A Complex Variable ODE

suppose $f$ is a holomorphic function on some domain $D$ satisfying $f'(z)=af(z) $ for some >constant a. show that $f(z)=Ce^{az}$, for some constant $C$
1
vote
4answers
191 views

“philosophical” question about the transcendence of $\pi$

I don't have any knowledge on transcendence proofs. I just heard that Lindemann proved that for any $\alpha \in \mathbb R^*$ algebraic, $e^\alpha$ is transcendental. Then, since $i$ is algebraic, and ...
1
vote
2answers
58 views

Finding power series

I need to find the power series for $e^z + e^{az} + e^{a²z}$ where $a$ is the complex number $e^{2πi/3}$. I know that $1 + a + a² = 0$. I have tried to differentiate the expression and give values ...
4
votes
2answers
123 views

If $f$ is entire and $\exp(f(z))$ is a polynomial, then $f$ is constant.

In a recent question that was just deleted, @danielfischer gave at the end of his answer the following exercise: for entire $f$, $$e^{f(z)} \text{ is a polynomial} \iff f \text{ is constant}$$ I was ...
2
votes
1answer
89 views

$z\exp(z)$ surjectivity with the Little Picard Theorem

I would like to prove the surjectivity of this function : \begin{align*} f\colon\mathbb{C}&\to\mathbb{C}\\ z&\mapsto z\exp(z) \end{align*} You can use the Little Picard Theorem: If a ...
1
vote
2answers
74 views

Real solutions of the equation $\,z + e^{-z} - x = 0$ for $\,x > 1$ and $\,\Re(z) ≥ 0$

I have the complex equation $z + e^{-z} - x = 0$, where $z$ is complex and $x$ real, and I need to show that it has only one real solution for $x > 1$ and $\Re(z) \ge 0$. I do not see how this is ...
2
votes
1answer
150 views

Proofs that there is no $f(z)$ such that $\exp f(z) = z$ for all $z \in \Bbb{C}\setminus\{0\}$

When I first learned about this result I was completely stunned that there is no holomorphic function $f(z)$ on $\Bbb{C}\setminus\{0\}$ such that $\exp f(z) = z$. What are some interesting proofs of ...
1
vote
1answer
375 views

Find the Maximum and Minimum values of $e^z$ when $z\le 1$.

I need help finding the maximum and minimum values of $|e^z|$ on $|z|\le1$. I know we use the maximum modulus theorom but i cant seem to get an answer.
7
votes
2answers
129 views

What are the properties of the roots of the incomplete/finite exponential series?

Playing around with the incomplete/finite exponential series $$f_N(x) := \sum_{k=0}^N \frac{z^k}{k!} \stackrel{N\to\infty}\longrightarrow e^z$$ for some values on alpha (e.g. ...
1
vote
0answers
62 views

How to analyze the asymptotic properties of this function?

Let the function $$f(\mathbf{r})=\int_{\Omega }e^{i\mathbf{k} \cdot \mathbf{r}}d^2\mathbf{k}$$, where $\mathbf{k} ,\mathbf{r}\in\mathbb{R}^2$, and $\Omega \subset \mathbb{R}^2$ is some finite region ...
2
votes
2answers
96 views

Evaluating $\int _{-1}^{e} \frac{1}{x}dx$

Here very easily by the Fundamental Theorem of Calculus $$\int _{-1}^{e} \frac{1}{x}dx=\ln(e)-\ln(-1)$$ From Euler's identity $e^{i \pi}$=-1 we can easily deduce that $\ln(-1)=i \pi$. Thus the ...
0
votes
0answers
50 views

splitting up summations of product

I want to check the validity of splitting up summations of products. I am using the DFT matrix and am trying to get a simplified expression of it . In essence, I am trying to prove the following lemma ...
1
vote
1answer
49 views

Expressing a function as exponential of another function in a given domain

Can the functions $h_1(z) = 1+z^2$ and $h_2(z) = 1+\log(4+z)$ be expressed as the exponential of a function $f(z)$, where $f$ is holomorphic in the set $D = \{z : |z| < 2 \}$ ? More generally, ...
1
vote
0answers
85 views

Exponential functions of a complex number. I can't get it right.

Question: Prove that $\frac{e^Z}{e^W} = e^{z-w}$ My Attempt: The given equation on left-hand side can be re-written as $e^z.e^{-w}$. Let $z = x + iy$ and $w = u + iv$. $exp(w) = e^u(cosv+isinv)$. ...
4
votes
4answers
89 views

Proof of $e^z \neq 0$

Proof: Let $a \in \mathbb{C}$ be s.t. $e^{a} =0$. Then $0=e^a e^{-a} = e^{-a+a} = e^0 =1$, contradicting the existence of $a$. But why can we multiply by $e^{-a}$?? If $e^a=0$, then ...
1
vote
1answer
66 views

Proving that $\exp(z_1+z_2) = \exp(z_1)\exp(z_2)$ with power series [duplicate]

Probably a simple question, but I wonder about the following: To prove that $\exp(z_1+z_2) = \exp(z_1)\exp(z_2)$, I use : $$\exp(z_1+z_2) = ...
4
votes
1answer
188 views

Solving a transcendental equation consisting of a quadratic part and a part involving inverse Lambert W functions

Question statement I would like to solve the following equation in the two variables $x$ and $y$: \begin{gather} 0 = x^2 - a y^2 + i b [x y - W^{-1}(x)W^{-1}(y)] , \end{gather} where $a$ and $b$ are ...
3
votes
6answers
117 views

Motivation for creation of complex exponentiation

I am curious how mathematicians came to develop complex exponentiation. How is the rule for complex exponentiation derived?
2
votes
1answer
88 views

Why is $e^{g(x)} = \pi$ where $g(x)$ is holomorphic in Weierstrass factorization of sine function?

Why is $e^{g(x)} = \pi$ where $g(x)$ is holomorphic in Weierstrass factorization of sine function? I just can't get why it's true.
2
votes
3answers
163 views

Incoherence using Euler's formula

Using the relation $\ e^{ix} = \cos(x) + i\sin(x)$ and substituting for $\ x = \pi$, we have the well-known Euler identity, $ e^{i\pi} = -1$. Substitute also for $ x = -\pi $, we have $ e^{-i\pi} = ...
1
vote
1answer
155 views

Connected set on complex plane

What's the numebr of connected components for the set of complex numbers $\{e^z:|z|=1\}$ on the complex plane? Remark: It represents a simple closed curve which intersects the real axis at points ...
3
votes
2answers
79 views

Complex representation of sinusoids question

A sum of sinusoids defined as: $$\tag1f(t) = \sum_{n=1}^{N}A\sin(2\pi tn) + B\cos(2\pi tn)$$ is said to be represented as: $$\tag2f(t) = \sum_{n=-N}^{N}C\cdot e^{i2\pi tn}$$ which is derived from ...
1
vote
2answers
276 views

Entire function which equals exponential on real axis

I need to find all entire functions $f$ such that $f(x) = e^x$ on $\mathbb{R}$. At first it seems that, since the function $f$ will be real analytic on $\mathbb{R}$ and will have a power series ...
1
vote
1answer
994 views

Radius of convergence for the exponential function

I'm studying physics and am currently following a course on complex analysis and in the section on analytic functions, the radius of convergence $R$ for power series was introduced. The Taylor ...
1
vote
2answers
255 views

branch of logarithm

What's the shortest way to show that there is no analytic function $f$ on $\mathbb{C} \backslash \lbrace 0 \rbrace$ such that $$\exp(f (z)) = z$$ for all nonzero complex numbers $z$? I came across ...
1
vote
2answers
185 views

Find the limit of $\frac{\bar{z}}{z}$ as $z$ goes to $0$.

I put it in exponential form to get $\dfrac{re^{-i \theta}}{re^{i \theta}}$ but I think I'll get $\frac{0}{0}$ which isn't defined and isn't a good enough proof to say it doesn't have a limit.
0
votes
0answers
56 views

Question about elementary and nonelementary functions.

Let $E(z)$ be an entire elementary function of (complex) $z$ and $N(z)$ be an entire nonelementary function of (complex) $z$. $e^{N(z)}$$N'(z) = E(z)$ The ' means derivative with respect to $z$. ...
-1
votes
1answer
175 views

How to find asymptotic entire functions?

I want to know how to find analytic functions $f(z)$ that are asymptotic and analytic on and near the real line of functions of the type $\ln(C +\exp(P(z^2)))$ where $C$ is a complex constant and $P$ ...
2
votes
2answers
260 views

Exponential function and bijection

I am required to prove the following: For any real number $k$, prove that the exponential function $e^z$ is a bijection ($z$ is a complex number) from the strip $a < im z \leq k+2pi$ to the ...