Recent content by babyrudin

Any idea on this?

Great, I think I know how to do it now. I was trying the Cauchy integral formula too much. Thanks!

I've got it now, many many thanks!

Homework Statement For x,y \in R^n, their inner ("dot") product is given by <x,y>=\sum_{i=1}^n x_i y_i. Also, we write <x,x>=\|x\|^2. Homework Equations Fix p>1. Show that for all x,y \in R^n we have < \|x\|^{p-2}x -\|y\|^{p-2}y, x-y> \geq 0The Attempt at a Solution Expanding the...

Homework Equations Let w \in C be a fixed complex number with |w|<1. Let f(z)=\frac{z-w}{1-\bar{w}z}. Calculate its maximum value in the region |z| \leq 1. The Attempt at a Solution How should I approach this? Not sure if maximum modulus principle is of much help.

Homework Equations Using complex analysis, compute \int_{-\infty}^{\infty} \frac{e^{itx}}{1+x^2}dx where t is real. The Attempt at a Solution I'm not good at complex analysis at all and am totally lost. I do know some Fourier analysis though and using it I got \pi e^{-|t|}. How should I...

Homework Statement Let f(x) be an integrable complex-valued function on \mathbb{R}. We define the Fourier transform \phi=\mathcal{F}f by \[\phi(t)=\int_{\infty}^{\infty} e^{ixt} f(x) dx.\] Show that if f is continuous and if $\phi$ is integrable, then \[f(x)=\frac{1}{2\pi}...

Hello all! In solving some math problems, I encountered the following sum: \sum_{k=1}^{r+1} kb \frac{r!}{(r-k+1)!} \frac{(b+r-k)!}{(b+r)!}. \quad \mbox{(eqn.1)} Now, I have asked Maple to calculate the above sum for me, and the answer takes a very simple form: \frac{b+r+1}{b+1}. \quad...