1
vote
1answer
54 views

Definition of $z^0, z=a+bi, a,b \in \mathbb{R}, z \neq 0$

Today I found in a True-False the question; Does the equality $$z^0=1, z=a+bi, a,b \in \mathbb{R}$$ hold $\forall z \in \mathbb{C^*}$? The thing is, this was never clearly defined in the book, and ...
0
votes
3answers
133 views

How is $ i^{-1} = -i$ and $i^{-3} = i$?

Now I know that with positive powers of $i$ the cycle is: $i , -1 , -i , 1\ldots$ The negative power cycle is: $-i , -1 , i , 1 \ldots$ Can someone explain to me how $\frac 1 {\sqrt{-1}}$ is equal ...
4
votes
0answers
64 views

Definition of $0^z$, $z$ complex

Usually we define the function $a^b$ over the complexes using the exponential function, as $e^{b\log a}$. This function has some issues with multivalued-ness, but it still more or less satisfies ...
2
votes
4answers
119 views

$(-1)^{0.2}=0.8090 + 0.5878i$ how can this be?

I'm working on a numerical analysis project (working with matlab a lot) and I noticed that when I ask for matlab to compute the exponent of a negative number, it gives wrong output when the exponent ...
0
votes
1answer
41 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
47 views

Writing $e^{i\theta}(e^{in\theta}-1)/(e^{i\theta}-1)$ in $(a+i b)$ form

How to write: $$e^{i\theta}\cdot\frac{e^{in\theta}-1} {e^{i\theta}-1}$$ in $$(a+i b) $$ $$ ?$$ I tried to multiplicate by $$e^{i}$$ (the numerator and ...
4
votes
2answers
117 views

An “elementary” approach to complex exponents?

Is there any way to extend the elementary definition of powers to the case of complex numbers? By "elementary" I am referring to the definition based on $$a^n=\underbrace{a\cdot a\cdots ...
0
votes
1answer
61 views

A complex series with exponentials

I have tried to solve this type of series : $$\sum \frac{e^{i\, u(n)}}{v(n)} $$ For some $u,v$ an Abel Transform allow to find convergence, but for $u(n)=n^2$ and $v(n)=n$ I can't find an argument. ...
0
votes
2answers
40 views

Simplifying $\exp {- i 2 \pi / N}$.

a and b are complex numbers and I know the equation below. $$X_{N} = a + e^{-i2\pi /N}*b$$ I wanted to simplify it. Here is what I've tried. I know $e^{-i\pi} = -1$ $X_{N} = a + \left ( e^{i\pi} ...
1
vote
1answer
65 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 ...
0
votes
3answers
49 views

Error raising a complex number to a power

I am trying to do $(3+7i)^5$ which acording to WolframAlpha and Mathway should be: $23028−11228i$ Yet I instead get: $6123+14287i$ -- I'm getting that answer by doing: $3^5 ...
2
votes
2answers
48 views

magnitude of complex exponental always equals 1?

as we all know $$e^{j\theta} = \cos\theta + j\sin\theta \\ |e^{j\theta}| = \sqrt{\cos^2\theta + \sin^2\theta} = 1$$ That means $|e^{j\theta}| = 1$ with any value $\theta$ is ($2\pi, \frac{\pi}{3}$, ...
0
votes
2answers
53 views

complex expression to the power of a complex expression

I have a math exam tomorrow, and i am not sure with my solution for a exercise. can you please tell me if i am right. Question is: $$(1+i)^{(1-i)}$$ My solution is: $$\sqrt{2} e^{(i ...
4
votes
2answers
99 views

How to resolve a power of a negative number?

$\left(-64\right)^{\left(\frac{3}{2}\right)}$ (Disclaimer - I work in a HS math center, helping students. This is from an Algebra/Trig text used by both sophomores and juniors depending on the class. ...
0
votes
1answer
65 views

Is $\left((-1)^2\right)^\frac12 = (-1)^\left(2\cdot\frac12\right)$? [duplicate]

I'm feeling confused. If I square 1 and -1, the answers should be equal: $1^2 = (-1)^2$ Then I take both sides to the power of $\frac12$: $\left(1^2\right)^\frac12 = \left((-1)^2\right)^\frac12$ ...
2
votes
0answers
44 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 ...
0
votes
1answer
114 views

How do I solve $|\sinh(x+iy)|^2 = (\sin(y))^2+(\sinh(x))^2$

How do I solve this ? $|\sinh (x+iy)|^2 = ( \sin (y))^2+ ( \sinh (x))^2$ I'm not sure how to solve the left hand side.
1
vote
0answers
41 views

Rules for $(x^a)^b$ - complex number. [duplicate]

How potentiate $(x^a)^b$ for complex numbers?
0
votes
3answers
80 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?
5
votes
2answers
390 views

Non-integral powers of a matrix

Question Given a square complex matrix $A$, what ways are there to define and compute $A^p$ for non-integral scalar exponents $p\in\mathbb R$, and for what matrices do they work? My thoughts ...
6
votes
6answers
220 views

What is $(-1)^{\frac{2}{3}}$?

Following from this question, I came up with another interesting question: What is $(-1)^{\frac{2}{3}}$? Wolfram alpha says it equals to some weird complex number (-0.5 +0.866... i), but when I try ...
1
vote
2answers
53 views

Polar form for complex number with variable exponent

This might be an easy question but I'm having trouble showing that $\left(1+\frac{i\theta}{m}\right)^m$ has the angle $ m\arctan\left(\frac{\theta}{m}\right)$ in polar form on the complex plane. ...
3
votes
3answers
91 views

Exponentials in complex numbers

If $\displaystyle z-\frac1z=i$, then find $\displaystyle z^{2014}+\frac{1}{z^{2014}}$. The answer should be in terms of $1, -1,\;i\;or\;-i$. I am not able to understand how to simplify the given ...
1
vote
2answers
199 views

What does $i^i $ equal and why? [duplicate]

I've been reading up on why the value of 0^0 is controversial (see Zero to the zero power - Is $0^0=1$?) and I wondered: is it possible for $i^i$ to have a value? I plugged it into a TI-83 calculator ...
0
votes
0answers
95 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 ...
2
votes
4answers
250 views

Raising a Complex Number to a Decimal Value

So for my class i have to make a java program that deals with complex numbers. I finished getting the root and power and i was wondering how to do a method that deals with powers such as 2.56. Now im ...
0
votes
2answers
142 views

Rational exponents: prove some states

In some rational exponent expressions the solution isn't a real number why? Example (explain what I mean): $$\begin{align} \Big(-x\Big)^{1/n}=\left\{\text{is not a real number}\right\} \end{align}$$ ...
7
votes
2answers
301 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 ...
0
votes
3answers
95 views

Can $(-1)^{a+i b}$ be expressed without negative based exponentiation, complex exponentiation, complex logarithms or trigonometric functions?

Can this expression, where $a$ and $b$ are both real, be expressed without negative based exponentiation (i.e. $a^b$ where $a$ is negative), complex exponentiation, complex logarithms or trigonometric ...
3
votes
3answers
178 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 ...
1
vote
2answers
87 views

Do any issues arise if we try to raise an element of $\mathbb{R}^+$ to an element of $\mathbb{C}$?

If $a$ and $b$ are non-zero natural numbers, the definition of $a^b$ is clear. Now it seems to me that there are (at least) two distinct ways of generalizing to larger number systems. Firstly, given ...
0
votes
1answer
67 views

Why $\frac{1}{2i}(e^{i\omega t} - e^{-i\omega t}) = \frac{i}{2} (e^{-i \omega t} - e^{i\omega t})$

Let $i := \sqrt{-1}$, $f$ be the frequency ($\frac1p$), and $\omega := 2 \pi f$. From page 3 here, why does $\frac{1}{2i}(e^{i\omega t} - e^{-i\omega t}) = \frac{i}{2} (e^{-i \omega t} - e^{i\omega ...
1
vote
1answer
51 views

Can we write $\sqrt[w]{z}=z^\frac{1}{w}$ when both $w$ and $z$ are complex numbers? [duplicate]

Let $w$ and $z$ be complex numbers defined in terms of real numbers $a$, $b$, $c$ and $d$ as follows: $$ w = a+bi \\ z = c+di $$ Can we analogically write $$ \sqrt[w]{z} = z^\frac{1}{w} \qquad ...
1
vote
4answers
131 views

Complex power of a complex number: Find $x$ and $y$ in $x + yi = (a + bi)^{c+di}$

$$ x + yi = (a + bi)^{c+di} $$ Find $x$ and $y$ in terms of $a$, $b$, $c$ and $d$. Where, $i$ is defined as $\sqrt{-1}$ and $a$, $b$, $c$, $d$ are real numbers. I defined two new real number ...
0
votes
1answer
61 views

Put the following in rectangular form.

$$(\sqrt{3}+i)^7$$ My question: $r = 2$. For $\theta$, do I use $\dfrac{\pi}{6}$ or $\dfrac{\pi}{6} + 2n\pi$? The book uses the former but I thought the latter is more appropriate. Thank you.
0
votes
0answers
34 views

check validity of following manipulation

in my algebra book,there is written following well known identity $e^{2*\pi*i}=1$ generally we can use also this identity $e^{k*\pi*i}=(-1)^k$ and if instead of $k$,we put $2$ we get ...
27
votes
14answers
2k views

How do I understand $e^i$ which is so common?

Raising something to an imaginary number is weird, I have a hard time wrapping my head around that. And e seems even more common and comes up in many situations, such as: the non-geometric ...
2
votes
4answers
242 views

What is the value of $i^i$? [duplicate]

I understand that when you raise any number $x$ to a power, you multiply $x$ by itself the number of times indicated in the power. However, what happens when $i^i$ is performed? How can a number be ...
4
votes
3answers
282 views

Where is the mistake in this proof?

I can't figure out, where is the mistake: $$z=re^{i\phi}=re^{\large \frac{2\pi i\phi}{2\pi}}=r(e^{2\pi i})^{\large\frac{\phi}{2\pi}}=r1^{\large\frac{\phi}{2\pi}}=r1=r$$ And we found that the complex ...
5
votes
7answers
466 views

Is $0^0=1$ postulate independent of all other axioms of complex numbers?

This question is inspired by the other question which asked for a proof that $i^i$ is a real number. Many calculators when asked for $0^0$ return 1. I asked a mathematician how to prove that but he ...
4
votes
4answers
187 views

$(1+i)$ to the power $n$ [duplicate]

Possible Duplicate: Complex number: calculate $(1 + i)^n$. I came across a difficult problem which I would like to ask you about: Compute $ (1+i)^n $ for $ n \in \mathbb{Z}$ My ideas so ...
25
votes
1answer
355 views

Iterated exponent of $i$

WolframAlpha seems to tell me that $e^{e^{e^{e^{e^{e^{e^{e^{e^{e^{e^i}}}}}}}}}} = 1$, see link. Is this just an error or is it for real? Adding one more $e$ to the bottom of the tower gives me the ...
0
votes
3answers
539 views

How complex exponential converges and “sum of exponents” rule holds

How is it the complex exponential converges for any value of $z$ in the complex plane? $$e^{z} = 1 + \frac{z}{1!} + \frac{z^2}{2!} \cdots\cdots$$ How is it the "sum of exponents" rule holds for ...
4
votes
2answers
146 views

Is the multiplicative complex plane a Lie group?

I know that the complex plane is a Lie group with +, but is it also a Lie group with the usual complex multiplication? This would give us a nice geometrical interpretation of the famous Euler ...
3
votes
2answers
217 views

How to find complex numbers $z,\lambda,\mu$ such that $(z^\lambda)^\mu\neq z^{\lambda\mu}$

Let $z$, $\lambda$, $\mu$ be complex numbers. Find a case where $(z^\lambda)^\mu$ is not equal to $z^{\lambda\mu}$. In our book, $a^b = \exp( b \cdot \operatorname{Log}(a) )$. ...
1
vote
2answers
319 views

Fixed points of $e^z$

How would one find the fixed points of $e^z$, where $z$ is complex (if there are any)? I feel this problem probably has a really obvious answer, and for some reason, I'm just not getting it. Thanks.
-2
votes
1answer
283 views

Smallest positive integer for equation

I am having trouble identify the smallest positive integer $n$ such that $(\frac{1+i}{1-i})^n = 1$ Can someone please throw on approach? (Also, please correct the equation in the form of Tex/Latex ...
1
vote
1answer
191 views

Are numbers : $(-1)^{i} , 1^{-i} , 1^{i} $ transcendental numbers? [duplicate]

Possible Duplicate: What is the value of 1^i? According to Euler's formula : $e^{ix}=\cos x + i\cdot \sin x$ we may write : $$e^{i\cdot \frac{\pi}{2}}=i \Rightarrow \left(e^{i\cdot ...
45
votes
4answers
2k views

A new imaginary number? $x^c = -x$

Being young, I don't have much experience with imaginary numbers outside of the basic usages of $i$. As I was sitting in my high school math class doing logs, I had an idea of something that would ...