# Tagged Questions

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### Branch of $n$th root of $f$ is holomorphic

The problem states to prove that if $h$ is a branch of $f^{1/n}$ for integer $n > 0$ (i.e. $h(z)^n = f(z)$ for $z \in G$, $h$ continuous), then $h$ is holomorphic, where $f$ is a holomorphic ...
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### 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, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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, ...
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### 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 ...
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### 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.] $(i)$ ...
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### 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 ...
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### 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 $C$
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### “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 ...
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### Finding power series

I need to find the power series for $e^z + e^{az} + e^{aĀ²z}$ where $a$ is the complex number $e^{2Ļi/3}$. I know that $1 + a + aĀ² = 0$. I have tried to differentiate the expression and give values ...
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### 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 ...
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### $z\exp(z)$ 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 ...
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### Real solutions of the equation $\,z + e^{-z} - x = 0$ for $\,x > 1$ and $\,\Re(z) ā„ 0$

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 ...
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### 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 ...
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### 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.
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### 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. ...
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### 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 ...
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### 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 ...
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### 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, ...
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### 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 $w = u + iv$. $exp(w) = e^u(cosv+isinv)$. ...
### 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 ...