2
votes
1answer
42 views

Convergence of an infinite power

There are complex numbers $z$ and $w$ for which $$\lim_{n\rightarrow\infty}z\uparrow\uparrow n=w$$ where $\uparrow\uparrow$ is the tetration symbol, e.g. $z=\sqrt{2}$ and $w=2$. Are there complex ...
1
vote
1answer
49 views

Why do we define the complex exponential as we do?

Why do we define the complex exponential as we do? Defining $e^{z}$ as $e^{x}e^{iy}$ certainly seems to make sense, but I'm not sure the formal reason as to why it's defined like this. Was it from ...
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 ...
1
vote
2answers
52 views

Definition of $a^b$ for complex numbers

Problem statement Let $\Omega \subset C^*$ open and let $f:\Omega \to \mathbb C$ be a branch of logarithm, $b \in \mathbb C$, $a \in \Omega$. We define $a^b=e^{bf(a)}.$ $(i)$ Verify that if $b \in ...
1
vote
0answers
60 views

If $f$ is holomorphic, then there is a holomorphic function $h$ such that $e^{h(z)}=f(z)$

Let $f:G\to\mathbb{C}$ denote a holomorphic function over a star-shaped domain $G$ and $f\ne 0$ on $G$. I want to show that it holds $\frac{f'}{f}$ is holomorphic There is a holomorphic function ...
1
vote
1answer
64 views

Conditions required for $(z_{1}z_{2})^{\omega}=z_{1}^{\omega}z_{2}^{\omega}$, where $z_{1},z_{2},\omega\in\mathbb{C}$

I am having trouble finding the conditions on $z_{1}$ and $z_{2}$ in order for: $$(z_{1}z_{2})^{\omega}\equiv z_{1}^{\omega}z_{2}^{\omega}\qquad \forall\omega\in\mathbb{C}$$ My first step was to ...
2
votes
2answers
66 views

compute the integral using residue theory

I am trying to compute an integral in an example in my complex analysis textbook: $$\int_{-\infty}^\infty {xsinx\over x^4+1}dx$$ The book gives some startup hints, but I don't quite follow, I set ...
0
votes
0answers
30 views

What is the general notation for the principal value of complex exponential?

It is general to distinguish the principal value of complex logarithm set by denoting it $Ln( z)$. Is there any general notation to distinguish the principal value of complex exponential? In complex ...
1
vote
1answer
20 views

Supremum of (e^(i z t) - 1)/z

i'm new here, so i'm not sure if this is the right place to ask this question: I know that the following holds true: $$ \forall\, t \in \mathbb{R} \; \forall\,x\in\mathbb{R}\setminus\{0\} ...
1
vote
1answer
56 views

Cauchy's integral formula used on circle

If $\gamma$ is a piecewise, smooth, positively oriented simple closed curve in $D$, then Cauchy's formula states that $f(z)=1/2\pi i\int_\gamma {f(a)\over {a-z}}$. My textbook also stated that for ...
13
votes
5answers
431 views

Value of $(-1)^x$ for $x$ irrational

I was working on an analysis problem when this question arose in one my proofs. I think it may be either $-1$ or $1$, but it seems like there can only be an arbitrary way to assign this. So is there ...
0
votes
0answers
67 views

Complex exponentiation

So I've got this question that is a bit difficult to ask, since it uses a term in my language that I can't properly translate into English. For $z\in\mathbb{C}^*$ and $a\in\mathbb{C}$ it would be ...
0
votes
1answer
43 views

Exponential of $\bar{z} $

I am currently reading the book Complex Variables by Stephen Fisher, there is one paragraph that was written like this: Establishing the following relation, and they write ...
2
votes
0answers
43 views

Find a holomorphic function that satisfies this equation

Let $f:\mathbb{C}\times\mathbb{C}\rightarrow\mathbb{C}$ denote a function that satisfies the following equation: $$f(w,z+1)=w^{f(w,z)}$$ Is there a unique holomorphic function $f$ that satisfies this ...
5
votes
2answers
125 views

How many values does $1^{\alpha}$ have for $\alpha$ irrational?

One such value is $\displaystyle\cos\left(2\pi\alpha\right)+i\sin\left(2\pi\alpha\right)$, by Euler's theorem. On the other hand, we can choose an arbitrary sequence $S=(a_n)_n$ of rational numbers ...
7
votes
2answers
292 views

Why is $\left(e^{2\pi i}\right)^i \neq e^{-2 \pi}$?

Here's my (obviously flawed) proof that $1=e^{-2 \pi}$: $$ 1^i=1\\ e^{2 \pi i} = 1\\ \left(e^{2\pi i}\right)^i = 1^i\\ e^{-2 \pi} = 1 $$ What's the issue? I understand that exponentiation is not ...
2
votes
2answers
198 views

Complex Exponent of Complex Numbers

How does one find the algebraic solution of a Complex number raised to the power of another Complex number? Here is the work I have done so far, if there are any mistakes please inform me. A real ...
3
votes
3answers
176 views

Can we *ever* use certain log/exp identities in the complex case?

This article on Wikipedia points out that certain identities for the log and exponential functions which are familiar from the real case require care when used in the complex case. Failures in the ...
6
votes
3answers
215 views

If $\theta\in\mathbb{Q}$, is it true that $(\cos \theta + i \sin \theta)^\alpha = \cos(\alpha\theta) + i \sin(\alpha\theta)$?

Is the following true if $\theta\in\mathbb{Q}$? $$(\cos \theta + i \sin \theta)^\alpha = \cos(\alpha\theta) + i \sin(\alpha\theta)$$ Is it true if $\alpha\in\mathbb{R}$? In each case, prove or give a ...
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?
1
vote
2answers
48 views

Calculating z^n without trigonometric functions

I'm looking for a formula to calculate z^n for complex z and integer n. I know that I can use the Moivre's formula but I will use the formula in a computer program ...
1
vote
1answer
126 views

How to expand a fraction in powers of $z$ or $\dfrac{1}{z}$, and which to do, in determining Laurent series

I have a function $f(z)=\dfrac{12}{z(2-z)(1+z)}$, I'm trying to find the Laurent series for each of the three annuli. The singularities are at $z = 0$, $z = 2$, and $z = -1$, so I'm looking for three ...
-4
votes
1answer
118 views

Boundedness on strips in the complex plane for functional equations [closed]

We know that the recurrence for $b>0$ (1) $f(0)=1$ (2) $f(z+1)=b{f(z)}$ has $f(z)=b^z$ as the only entire solution that is bounded on the strip $S=\{z: 0<\Re(z)\le 1\}$. The image of $S$ ...
3
votes
0answers
166 views

What is $i^{i^{\,{}^{.\,^{.\,^{.\,^i}}}}}$ equal to? [duplicate]

Possible Duplicate: How to compute the infinite tower of the complex number $i$, that is$ ^{\infty}i$ Define $a_0 = i$, and $a_n = i^{a_{n-1}}$. What is $$\lim_{n \rightarrow ...
1
vote
2answers
741 views

Find all solutions to the equation $e^z = i$

I know that there is an equation for finding the nth roots of a complex number, which easily done once you have the modulus and argument of the complex number in question. There would be n roots. But ...
2
votes
2answers
75 views

Simplifying this exponential equation

I am wondering how does $$\frac{{{e^{zk}}}} {{{z^2} + 1}} = \frac{1} {{2i}}\left( {\frac{{{e^{zk}}}} {{z - i}} - \frac{{{e^{zk}}}} {{z + i}}} \right)?$$ I can see that $z^2 + 1 = (z + i)(z − ...
1
vote
1answer
112 views

Branching of $z^z$

I'm facing the problem of proving that ($0,+\infty)$ is a branching point of $f(z)=z^z$. Now, straight from the definition, if $z=|z|e^{i\theta}$ with $0 \le \theta < 2\pi$ $$ z^z := e^{z\log z} = ...
1
vote
1answer
204 views

Limit of a power function with imaginary power

Is there is any precise notion of the limit $\lim_{x\rightarrow0^+}x^\mu$ where $\mu$ is purely imaginary?
8
votes
3answers
396 views

Proof for law of complex exponents using only differential equation

I just read that an elegant proof exists that the law of exponents also holds for complex numbers ($a,b,z$ all complex): $$e^{a+b}=e^ae^b,$$ which only uses the definition that $$y=e^{zt}$$ is a ...
5
votes
2answers
191 views

function $(-2)^{x}$

What is the real and imaginary parts for the function $f(x)=(-2)^{x}$ ? Is there a unique solution to this question?
2
votes
1answer
553 views

Is the local inverse of an analytical function locally analytical as well?

Take for example $f(z)=e^z$, so the inverse is $z(f) = \ln(f) + n\pi i$ for an arbitrarily chosen (but fixed) branch $n\in\mathbb N$. Now if $f$ is restricted to e.g. $0<|1-f|<1$ such that the ...