tonan6e

2022-10-07

Prove for any real number k, prove that the exponential function ${e}^{z}$ is a bijection (z is a complex number) from the strip $k<Im(z)\le k+2\pi $ to the complex plane minus the point 0, $\mathbb{C}-\{0\}$

Gabriella Hensley

Beginner2022-10-08Added 6 answers

Hint To solve ${e}^{x+iy}=\omega $, write $\omega $ in trigonometric form and solve.

KesseTher12

Beginner2022-10-09Added 6 answers

A hint: Don't solve equations, but investigate what the exponential function

$$z=x+iy\text{}\mapsto \text{}{e}^{z}={e}^{x}\cdot {e}^{iy}$$

does to horizontal lines

$${g}_{v}:\phantom{\rule{1em}{0ex}}y:=v\text{}(=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.)\text{},\phantom{\rule{1em}{0ex}}-\mathrm{\infty}x\mathrm{\infty}\text{},$$

and to vertical lines

$${h}_{u}:\phantom{\rule{1em}{0ex}}x:=u\text{}(=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.),\phantom{\rule{1em}{0ex}}-\mathrm{\infty}y\mathrm{\infty}\text{}.$$

$$z=x+iy\text{}\mapsto \text{}{e}^{z}={e}^{x}\cdot {e}^{iy}$$

does to horizontal lines

$${g}_{v}:\phantom{\rule{1em}{0ex}}y:=v\text{}(=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.)\text{},\phantom{\rule{1em}{0ex}}-\mathrm{\infty}x\mathrm{\infty}\text{},$$

and to vertical lines

$${h}_{u}:\phantom{\rule{1em}{0ex}}x:=u\text{}(=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.),\phantom{\rule{1em}{0ex}}-\mathrm{\infty}y\mathrm{\infty}\text{}.$$

Find the local maximum and minimum values and saddle points of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function

$f(x,y)={x}^{3}-6xy+8{y}^{3}$ $\frac{1}{\mathrm{sec}(x)}$ in derivative?

What is the derivative of $\mathrm{ln}(x+1)$?

What is the limit of $e}^{-x$ as $x\to \infty$?

What is the derivative of $f\left(x\right)={5}^{\mathrm{ln}x}$?

What is the derivative of $e}^{-2x$?

How to find $lim\frac{{e}^{t}-1}{t}$ as $t\to 0$ using l'Hospital's Rule?

What is the integral of $\sqrt{9-{x}^{2}}$?

What is the derivative of $f\left(x\right)=\mathrm{ln}\left[{x}^{9}{(x+3)}^{6}{({x}^{2}+7)}^{5}\right]$ ?

What Is the common difference or common ratio of the sequence 2, 5, 8, 11...?

How to find the derivative of $y={e}^{5x}$?

How to evaluate the limit $\frac{\mathrm{sin}\left(5x\right)}{x}$ as x approaches 0?

How to find derivatives of parametric functions?

What is the antiderivative of $-5{e}^{x-1}$?

How to evaluate: indefinite integral $\frac{1+x}{1+{x}^{2}}dx$?