Could somebody give me a good explanation for how $i^i$ works?  I'm a junior and just now getting to this.  I'm also too hard pressed for time to dive into exploring it myself.
2270
According to WolframAlpha,  but I don't know how I can prove it.
6953
Could someone please provide an explanation for the origin of ? (Ideally someone who I have been speaking with on the h-bar)  Any advanced maths beyond basic calculus will most probably leave me at a loss, though I do not mind a challenge! Please assume little prior knowledge, as over the past few days I have discovered that much of my understanding surrounding the process as virtual particle pairs is completely wrong.
68583
Where $i$ is the square root of negative one.  And is there a generalization of the $n$th root of $i$? Also how would this look graphically on the real number axis? Thanks
111357
Hello I don't understand meaning of the symbol represented by a capital letter "I" in the formulas below. There is  to the paper with the formulas.    At the beginning I thought it mean "where c is not equal to the result of classifier T(m)", however in step (d) there is a multiplication symbol before it, so it probably mean that "I" should "return" a value.
8078
We know $$i^2=-1 $$then why does this happen? $$ i^2 = \sqrt{-1}\times\sqrt{-1} $$ $$ =\sqrt{-1\times-1} $$ $$ =\sqrt{1} $$ $$ = 1 $$  EDIT: I see this has been dealt with before but at least with  I'm not making the fundamental mistake of assuming an incorrect definition of $i^2$.
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Should I write complex numbers as $a+ib$ or $a+bi$? Maybe there are certain cases where I should write it one way and other cases where I should write it the other way. What do you think?
59019
I am very confused about how to compute  $$\sum_{n=0}^{\infty}nk^n.$$  Can anybody help me?
4604
Why $0^0=I$?   I'd tried prove that considering $N^0$ where N is a Nilpotent matrix and then using the Cayley -Hamilton theorem  Thanks in advance.
12243
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 multiplied an imaginary amount of times? Wolfram Alpha says that it is equal to $e^{{-\pi}/{2}}$, but how would you arrive at that answer? Any response will be appreciated, thanks!
121569
From this paper:     Shouldn't $du$ be $dt$? And also how do you get from that line to the final result if $du$ is not $dt$?
71434
How can I show $i^{-1} = -i$, where $i$ is the imaginary unit?  Here's what I've tried: $i^{-1} = (-1)^{-1/2} = \dots ?$
12161
It's coming up to Christmas so I can ask to have all the books I can't afford from begrudging relatives! I'm really interested (mainly from looking at some of the answers cleo and other fantastic users!) in being able to approach integrals from a variety of different ways and learning how to tackle  non-elementary integrals.   I've gone over a lot of the standard techniques in my undergrad and this is just for a hobby, so don't want anything too 'heavy', just great explanations and a lot of questions to tackle. So far I've found Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals. Many thanks.
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I'm out of college, and trying to self-learn complex analysis. I'm finding  text difficult. Any recommendations? I'm probably at an intermediate sophistication level for an undergrad. (Bonus points if the text has a section on the Riemann Zeta function and the Prime Number Theorem.)
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Again, my instructor's explanation in class is so confusing to me.  Here we cannot take the original approach of finding the order and limiting our choices.  So instead my instructor says that this is isomorphic to $\mathbb{Z} \times \mathbb{Z} \times \mathbb{Z}_3$.  To justify it he said that $\mathbb{Z} \times \mathbb{Z} \times \mathbb{Z}$ is 3 dimensional and the denominator is one dimensional but this doesn't quite explain where the $\mathbb{Z}_3$ comes from?  Can anyone provide a better way to think about this?  It is making no sense to me at all.
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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.
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What would the steps be for someone new trying to learn how to solve this? I feel like I understand the concepts and what they mean but not how to prove it. Thanks
15155
So my friend and I are working on this and here is what we have so far.  We want to show that $\exists \, B$ s.t. $(I+A)B = I$. We considered the fact that $I - A^k = I$ for some positive $k$. Now, if $B = (I-A+A^2-A^3+ \cdots -A^{k-1})$, then $(I+A)B = I-A^k = I$. My question is: in matrix $B$, why is the sign for $A^{k-1}$ negative? Couldn't it be positive, in which case we'd get $(I+A)B = I + A^k$?  Thank you.