Difficult limits every grad should be able to do I'm trying to brush up on some advanced calc/ multivariable calculus stuff this summer and wanted to know what some of your professional opinions are on certain problems every 1st year grad should know in analysis. Difficult limits, integrals, proofs, etc that you deem as "need to know" or "pull out of your trick-pocket at any time". Maybe unconventional problems that are pretty neat? Thanks! 
 A: Perhaps the ones that are not so difficult but that rely on technical skills are more important for many purposes. But there are some special cases that should be known.
A few more-or-less random examples off the top of my head:


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*The Gamma function is $$\Gamma(\alpha) = \int_0^\infty x^{\alpha-1} e^{-x}\,dx.$$  How do you show that $\Gamma(\alpha+1)=\alpha\Gamma(\alpha)$?  Not at all difficult, but worth knowing. (Why does one use $\alpha-1$ instead of $\alpha$?  One reason is that if $f_\alpha(x)= x^{\alpha-1}e^{-x}$ for $x>0$ and $0$ for $x<0$ then $(f_\alpha * f_\beta)(x) = f_{\alpha+\beta}(x)$ (convolution).)  This comes up in probability theory.

*The Beta function is $$B(\alpha,\beta)=\int_0^1 x^{\alpha-1}(1-x)^{\beta-1}\, dx.$$ How do you show that $$B(\alpha,\beta)=\frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)}\text{ ?}$$ Not difficult, but worth knowing.

*$$\int_{-\infty}^\infty e^{-x^2/2}\,dx=\sqrt{2\pi}.$$ This matters in probability theory.  Polar coordinates are not the only way to do this, but that way works.


OK, I didn't really intend the pun in the line starting with "A few"${}\,\ldots$
I'm going to get lazy and leave this incomplete list here for now.  Maybe later I'll be back to extend it to a longer incomplete list.
Perhaps one should list techniques rather than particular integrals.
continued${}\,\ldots\,{}$:
Since you mention proofs:


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*How do you prove that $$f(a) = \frac 1 {2\pi i} \int_C \frac{f(z)}{z-a}\,dz,$$ if $f$ is holomorphic everywhere in the region surrounded by $C$, a curve that winds once counterclockwise around $a$?

*How do you use that to prove that if $f$ is holomorphic at $a$ (for now let's take that to mean differentiable everywhere in some disk of positive radius centered at $a$) then $f$ is analytic at $a$, i.e. $f(z)=$ the value at $z$ of some power series centered at $a$, which converges everywhere in that disk?

*How do you prove that $$e^z=\sum_{n=0}^\infty \frac {z^n} {n!} = \lim_{n\to\infty}\left(1 + \frac z n\right)^n\text{ ?}$$

*If you're going to teach, how do you explain to undergraduates what is "natural" about $e$? (Maybe the shortest answer is $(d/dx)a^x = (a^x\cdot\text{constant})$, and the "constant" is $1$ only if the base is $e$.)


The question seems to broad for any good answer, but maybe it's not if taken in its entirety as referring to what every good student at a certain point in his or her studenthood should know.  One can't experct omniscience.
Maybe more later${}\ldots$
