0
votes
1answer
38 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
1answer
57 views

Exponentiation of imaginary operator

It is very easy to prove that if $D=\dfrac{d}{dx}$, then $(e^{nD}f)(x)=f(x+n)$ about $x=m$ in the real numbers. Proof: $$(e^{mD}f)=\sum^\infty_{n=0}\dfrac{D^nf}{n!}m^n\\ \implies ...
0
votes
2answers
36 views

Logs of a complex number

Write a solution in Cartesian for of What should come next?
2
votes
2answers
58 views

Is this a valid proof for eulers formula?

I am wondering whether this proof is a valid proof of Eulers formula: $e^{ix}=i\sin(x)+\cos(x)$ $$\frac{d}{dx}e^{ix} = i(e^{ix})$$ $$\frac{d}{dx}(i\sin(x)+\cos(x)) = i\cos(x)-\sin(x) = ...
3
votes
4answers
288 views

Absolute value of complex exponential

Can something explain to me why the absolute value of a complex exponential is 1? (Or at least that's what my textbook is telling me.) For example: $$|e^{-2i}|=1, i=\sqrt {-1}$$
1
vote
1answer
43 views

Continuity and other properties of complex exponential

So I think I can do the others, but part (i) about showing the continuity of $a^z$ has me stumped. I always get really stuck when it comes to proving continuity (I am using the metric spaces ...
0
votes
1answer
83 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 ...
1
vote
2answers
57 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 ...
2
votes
2answers
70 views

Express $-1+i$ in exponential form.

Express $-1+i$ in exponential form. My attempt so far Let $z=-1+i$ $$r=|z|=\sqrt2$$ $$\theta=\tan^{-1}(-1)=-\frac{\pi}{4}$$ Now, this is where I go wrong (I don't know why it's wrong!): So in ...
0
votes
3answers
78 views

Complex exponential

I know that the equation $e^{z}=-1$ has no solution had if been $z$ is a real number. So does the equation also has no solution when $z$ is complex?
1
vote
2answers
66 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 ...
0
votes
4answers
168 views

$\lim _{ n\rightarrow \infty }{ { \left( -1+\frac { 1 }{ n } \right) }^{ n } } =? $

We know that $$\lim _{ n\rightarrow \infty }{ { \left( 1-\frac { 1 }{ n } \right) }^{ n } } =\frac { 1 }{ e } .$$ However the result of $$\lim _{ n\rightarrow \infty }{ { \left( -1+\frac { 1 }{ n ...
0
votes
0answers
90 views

Development of imaginary exponent without appealing to “ambiguity” between $i$ and $-i$

Is there a way to develop the definition of the imaginary exponent, $z^i$, for complex $z$, that does not appeal to the notion that $i$ and $-i$ are "qualitatively indistinct" and that does not rely ...
0
votes
2answers
119 views

Taking derivatives of exponential function

Beware, this question might be silly and may contain mathematical fallacies. $$ d/dt(e^{jwt}) = jwe^{jwt} $$ $$ d/dt(e^{j \pi t}) = j \pi e^{j \pi t} $$ $$ d/dt(e^{j 180 t}) = j 180 e^{j 180 t} $$ ...
12
votes
4answers
306 views

Euler's identity: why is the $e$ in $e^{ix}$? What if it were some other constant like $2^{ix}$?

$e^{ix}$ describes a unit circle in polar coordinates on the complex plane, where x is the angle (in radians) counterclockwise of the positive real axis. My intuition behind this is that ...
5
votes
1answer
159 views

Do there exist complex algebraic $α,β$ such that $α^β=π$ or $α^β=e$?

Given the algebraic operations and complex exponentiation $(a+bi)^{c+di}$ and logarithm, is it possible to derive $\pi$ and $e$? If one is derivable then so should be the other, as $e^\pi = ...
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 $w = u + iv$. $exp(w) = e^u(cosv+isinv)$. ...
0
votes
0answers
64 views

Problem of finding General value

The general value $e^i$ is given by___ $$e^i=e^{\cos(2n\pi+\pi/2)+i\sin(2n\pi+\pi/2)}=e^{e^{2n\pi+\pi/2}}, \quad \forall n\in I$$ Is it right? But here, I need answer $e^{-(2n\pi+\pi/2)}, \forall ...
1
vote
5answers
142 views

How does $Ae^{4ix}+Be^{-4ix}=A\cos(4x)+B\sin(4x)$?

$e^{ix}=\cos(x)+i\sin(x)$ $Ae^{4ix}=A(\cos(4x)+i\sin(4x))$ $Be^{-4ix}=B(-\cos(4x)-i\sin(4x))$ What am I doing wrong? I am trying to find the complimentary function of $\frac{d^2y}{dx^2} ...
7
votes
7answers
1k views

How to solve $e^{ix} = i$

I am taking an on-line course and the following homework problem was posed: $$e^{ix} = i$$ I have no idea how to solve this problem. I have never dealt with solving equations that have imaginary ...
1
vote
1answer
151 views

Logical explanation of Euler's formula

This question is a about (if not proving) at least guessing the Euler's formula. I don't want the proof using the infinite sums. We can guess by logic that for example that the equation ...
2
votes
3answers
154 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
151 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
0answers
96 views

Prove (*) by induction on k.

Challenge: For linear systems with constant coefficients, in some sense we "never need more" than the so-called exponential polynomials, meaning expressions in the form $$\sum_{i=1}^m ...
2
votes
5answers
564 views

How does $e^{i x}$ produce rotation around the imaginary unit circle?

Euler' formula states that: $e^{i x} = \cos(x) + i \sin(x)$ I can see from the MacLaurin Expansion that this is indeed true, however, I don't intuitively understand how raising $e^{i x}$ power ...
2
votes
1answer
568 views

Why aren't there more numbers like e, pi, and i? This is based on looking through the Handbook of Mathematical Functions and online.

This is kind of a big picture question. I just counted up all the symbols used in normal mathematics and, give or take, there are probably around 150 of them, tops. And that's really stretching ...
2
votes
2answers
255 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 ...
1
vote
1answer
141 views

Real, imaginary parts of $G(w)=\frac{1}{2}(\sqrt{2}-\sqrt{2}e^{iw})$

For the function $G(w)=\frac{1}{2}(\sqrt{2}-\sqrt{2}e^{iw})$ , show that; $\qquad\mathrm{Re}\,G(w)=\sqrt2\sin^2(w/2)\quad$ and $\quad\operatorname{Im}\,G(w)=-1/\sqrt2\sin w$. I really need help with ...
2
votes
1answer
4k views

Conjugate of exponential imaginary number

The conjugate of $e^{-iwt}$ is $e^{iwt}$. Then, what would be the conjugate of $e^{iwt}$? Would it be $e^{-iwt}$? Also, for $|e^{iwt}|^2$, what would the value look like?