0
votes
3answers
37 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
34 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
48 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
78 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
62 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
40 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
57 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
39 views

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

How potentiate $(x^a)^b$ for complex numbers?
0
votes
3answers
70 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
135 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
210 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
46 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
158 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
65 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
154 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
127 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
257 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
89 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 ...
2
votes
3answers
163 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
212 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
84 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
63 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
49 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
122 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
58 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
175 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
276 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
460 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
175 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 ...
23
votes
1answer
326 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
500 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 ...
2
votes
2answers
198 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
305 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
243 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
183 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 ...
43
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 ...
10
votes
4answers
516 views

Which step in this process allows me to erroneously conclude that $i = 1$

I was playing around with imaginary numbers and exponents and came up with this: $$ i = \sqrt{-1} $$ $$ \sqrt{-1} = (-1)^{1/2} $$ $$ (-1)^{1/2} = (-1)^{2/4} $$ $$ (-1)^{2/4} = ((-1)^{2})^{1/4} ...
1
vote
1answer
99 views

compute an exponential complex number

I have a pretty basic question about complex numbers. If $z=x+yi$, a complex number, i want to compute the real and the imaginary part of the number $w=e^{e^z}$. Thanks in advance for any help.
1
vote
3answers
309 views

A contradiction involving exponents

Where is the error in the following statement: $i^2=(i^2)^{\frac{4}{4}}=(i^4)^{\frac{2}{4}}=(1)^{\frac{1}{2}}=1$? I feel the error is in the first equality, because $(i^2)^{\frac{4}{4}}$ is in fact ...
8
votes
3answers
378 views

Proof for law of complex exponents using only differential equation

I just read that an elegant proof exists that the law of exponents also holds for complex numbers ($a,b,z$ all complex): $$e^{a+b}=e^ae^b,$$ which only uses the definition that $$y=e^{zt}$$ is a ...
5
votes
6answers
371 views

Can you explain $(1 + iX/n)^{n}$ without using e, sin, or cos?

In my ongoing strugle to understand $e^{\pi i}$ I managed to narrow down my conceptual difficulty. I'm having intuitive trouble understanding why $(1 + iX/n)^{n}$ is conceptually the same as a ...
3
votes
3answers
1k views

Complex Exponents

What does it mean to raise a number to a complex exponent, and why? A lot of the explanations that I've seen involve e, why is this? I'm looking for an intuitive answer describing to how ...
24
votes
3answers
1k views

What is the value of $1^i$?

What is the value of $1^i$? $\,$
6
votes
4answers
2k views

Non-integer powers of negative numbers

Roots behave strangely over complex numbers. Given this, how do non-integer powers behave over negative numbers? More specifically: Can we define fractional powers such as (-2)^-1.5? Can we define ...