What is $i^{1/i}$?  Step by step walk through would be appreciated.  Tried searching, but didn't find much about this.
3744
How do I calculate the outcome of taking one complex number to the power of another, ie  $\displaystyle {(a + bi)}^{(c + di)}$?
109742
I have been trying to prove this by induction on $n\in \mathbb{N}$, but this approach seemed to get me nowhere. I have a suspicion it might be necessary to express $\log{n}$ as $\int_1^n 1/x\text{ }dx$, but I could not build a rigorous argument from that.  Any help greatly appreciated!
40496
What's the thing with $\sqrt{-1} = i$? Do they really teach this in the US? It makes very little sense, because $-i$ is also a square root of $-1$, and the choice of which root to label as $i$ is arbitrary. So saying $\sqrt{-1} = i$ is plainly false!  So why do people say $\sqrt{-1} = i$? Is this how it's taught in the US?
31709
$A^2 + A + I= 0$   Can $A$ be singular? Justify your answer.  I do not know where to start.
74532
I am confused about something: \begin{eqnarray}     (e^{2 i \pi})^{0.5} = (e^{2 i \pi \cdot 0.5})= e^{i \pi}=-1 \end{eqnarray} but \begin{eqnarray}      e^{2 i \pi}=1~ and~ 1^{0.5}=1  \end{eqnarray} where is my mistake??  thanks
17686
In the example  $\cup^{i=1}_{\infty}F_i$   How do you get the i=1 directly below the union symbol and the infinity directly above?
52719
Possible Duplicate:          My calculator (as well as WolframAlpha) gives me the approximation:  $$0.2078795763507619085469...$$  But I don't understand how exponentiating two purely imaginary constructs yields a real (albeit irrational) number. When I do $i^{i+1}$ it gives me an imaginary number as well as $(i+1)^i$. So why does $i^i$ fall into that precise point where it is real and no longer imaginary? What is happening? I understand that exponentiation is not repeated multiplication, and it wouldn't make sense to multiply $i$ by itself $i$ times (because it would only yield $i$, $-i$, $1$, or $-1$). So what are we doing behind the scenes to get such a number?
27086
$$\sum_{n=1}^\infty \frac{i^n}{n}$$  I'm not sure how to approach this series to find if it converges/diverges because of the $i^n$. I tried using the comparison test comparing it with $\frac{1}{n}$ and the ratio test, but didn't get anywhere. How would you approach this and look at series of this form or how to look at $i^n$ in an infinite series?
110345
What is the value of $i^{i^{i^\ldots}}$?   My effort is the following: If $z, \alpha \in \mathbb{C}$ with $z \neq 0$ then we can write $z^{\alpha}=e^{\alpha \log z} = e^{\alpha [ \log |z|+i \text{ Arg z} + 2 \pi i m]}$ where $ m=0, \pm 1, \pm 2, \ldots$  and Arg z is the principle argument of the complex number $z$. But in above expression we don't have the finite no of exponents, so I am not sure how should I use this formula to the solve above problem. Thanks and Regards!
11328
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 exponential form: $-1+i=\sqrt2 e^{-i\pi/4}$  According to the solutions, $\theta=3\pi/4$.  Thanks in advance
5833
I simply have no idea how this result is found $$I=\hbar  \sqrt{\ell(\ell+1)}.$$ The result seems to just be dumped in textbooks rather than explained. I can get the result that $I_z=\hbar m_j$. Please explain as clearly as possible how this is derived.  Also, any pointers towards any books/videos that may help me learn quantum mechanics would be appreciated.
28652
If $X_1,...,X_n$ are iid, what is the mean of the following variable:?  $ \sum (X_i - \bar{X})^2$  I know the answer is $\sigma^2(n-1)$, but how is this calculated in general? If I expand, I get the variables squared, and that turns quite ugly. What's the trick here?
54597
How to find the real part of the complex number (in Euler's form) $ z = e^{e^{i \theta } } $ ?  I got confused on how to proceed.  I am a beginner to complex numbers.
63140
Following from  question, I came up with another interesting question: What is  $(-1)^{\frac{2}{3}}$?  Wolfram alpha  it equals to some weird complex number (-0.5 +0.866... i), but when I try I do this: $(-1)^{\frac{2}{3}}={((-1)^2)}^{\frac{1}{3}}=1^{\frac{1}{3}}=1$.If it has multiple "answers", should we even call it a "number"? Because if we don't, it would be a bit different from what we were taught in elementary school. I actually thought if it doesn't have a variable in it, it should be a number.  I'm a bit confused. Which one is correct and why? I would appreciate any help.
910
I stumbled across this relationship while I was messing around. What's the proof, and how do I understand it intuitively? It doesn't really make sense to me that the sum of odd numbers up to $2x + 1$ should equal $x^2$.
24958
I found$\ \prod_{i=1}^\infty \frac{p_i}{p_i-1}\approx 11.0453$ on . Moreover, writing a paper, should one provide a proof or it is trivial? Thanks in advance.
21095
I was hoping that someone could point me in the right direction on how I could start answering the following question:     Based on the fact that $\det(AB)=\det(A)\det(B)$ for all square $n \times n$ matrices $A,B$, explain in every detail why:   $$\det(A^{-1}) = 1/\det(A).$$