Questions tagged [complex-analysis]

Question 1

In Python or Sage, how do I test if an eisenstein integer is a primitive root modulo a complex prime over the ring? For instance, if I suspect (11 + 3√-3) is a ...

Question 2

Let $f$ and $g$ be entire functions such that $g(f(z)) = zf(z)$ for all $z \in \mathbb{C}$. What can you say about $f$ and $g$? I´ll write down what I have so far. My first attempt was differentiating ...

Question 3

If a real or complex function $f$ is undefined at a point $a$, but is sufficiently well-behaved near that point, we can find a sort of "average value" or "finite part" of $f$ at $a$...

Question 4

I work on the following equation $$ \frac{1}{z_1+z_3} = \frac{1}{z_1+z_2} + \frac{1}{z_1} + \frac{z_1}{z_2} + \frac{1}{(z_1+z_2)^2}. $$ Let $z_1$ and $z_2$ be given by $z_1=\frac{1}{2}\exp(i\alpha)$ ...

Question 5

I wish to ask a question on the Picard-Lefschetz method for computing conditionally convergent comlex integrals. There is a case in Picard-Lefschetz theory in which a steepest descent contour ...

Question 6

Consider the following integral: $$I=\int_0^\infty dx\,e^{-x^2\frac{1+j}{\sqrt{2}}}.$$ where j is the imaginary unit. We get: $$I^2=\int_0^\infty \int_0^\infty dx dy e^{-(x^2+y^2)\frac{1+j}{\sqrt{2}}}....

Question 7

e^iπ = -1 -e^iπ = 1 ln(-e^iπ) = Ln(1) = 0 ln(e^iπ) + iπ = 0 {ln(-x) = ln(x) + iπ} iπ.ln(e) + iπ = 0 2.iπ = 0 iπ = 0?

Question 8

Let $f$ be a conformal mapping from unit disk to a region G, say, $f(s)=z$. Then for any $s_1,s_2$ in the unit disk, how could the inequality below hold according to Distortion Theorem? $$|z_1-z_2|\ge\...

Question 9

Let $X$ be a compact real-analytic manifold. I know that by adding an extra condition to a Komatsu-Romieu sequence $\mathscr{M}$ one obtains the nonquasi-analytic class, and that this additional ...

Question 10

In my complex analysis book: Let $z_1, z_2, z_3 \in \mathbb{C}$ be three points in the complex plane. The triangle spanned by $z_1, z_2, z_3$ is the point set $$ \Delta:=\left\{z \in \mathbb{C} ; \...

Question 11

Let $f$ be a holomorphic function from an open subset of $\mathbb{C}$ to a complex Banach space. In this answer, it is proved that the Cauchy integral formula still holds, and it follows that $f$ is ...

Question 12

Let $f(z)$ be a meromorphic function on $\mathbb{C}$. Denote by $S$ the set of its zero's and poles. Let $\gamma$ be a (sufficiently smooth) closed curve in $\mathbb{C} \setminus S$. Is the following ...

Question 13

Let us consider the complex valued function $f(z)=\frac{1}{z}$ defined on the domain bounded a curve $\gamma(t)$, where $\gamma(t)=2+e^{it},~0 \leq t \leq 2 \pi$. $\gamma(t)=e^{it},~0 \leq t \leq 2 \...

Question 14

Let $ f(z)=\sum\limits_{n\ge 0} a_n z^n $ be a power series with radius of convergence $R>0$. Define $ S_0=\{z\in\mathbb{C}:|z|=R,\ \text{the series for } f(z)\ \text{converges}\}, $ and $ S_1=\{z\...

Question 15

Is a logarithm with base 1 defined in the field of complex numbers? I have not found any information about this. In real numbers, this is uncertain because $ \ln(1) = 0 $ and $ \log_a(b)= \frac {\ln(...

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Questions tagged [complex-analysis]

Testing Eisenstein Integer Primitive Root

$f$ and $g$ entire functions such that $g(f(z)) = zf(z)$

Is there a name for the limit $\lim_{\epsilon \to 0} \frac{f(a - \epsilon) + f(a + \epsilon)}{2}$?

Finding specific points on a complex parametric curve [closed]

Question on Stokes phenomenon in Picard-Lefschetz theory

On the branch selection for evaluating a Gaussian integral with complex exponent

Eulers formula I kinda find some errors mostly my mistake [duplicate]

Distortion Theorem on unit disk [closed]

Quasi-analytic classes are dense in its dual? Analytic functions are dense in the hyperfunctions?

How is he triangle spanned by $z_1, z_2, z_3$ is given by $ z=t_1 z_1+t_2 z_2+t_3 z_3, 0 \leq t_1, t_2, t_3, t_1+t_2+t_3=1$?

If $f$ is a holomorphic function to a Banach space whose image is contained in a dense subspace, is the image of $f'$ also contained in this subspace?

Continuous logarithm along closed path exists if and only if winding number along path is zero

What is the geometric intuition of complex integration and Cauchy theorem?

Relationship between boundary convergence of a power series and its derivative

Complex logarithm base 1

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