Recent content by Grothard

That's really cool; it does simplify the problem a lot. But I'm having a bit of trouble getting rid of that last term. I tried shifting the index once more, but that doesn't seem to help. It can't be factored out, and it's definitely nontrivial for a given term. Could you give me some tips as to...

Homework Statement Let f(z) = \sum_{n =-\infty}^{\infty} e^{2 \pi i n z} e^{- \pi n^2}. Show that f(z+i) = e^{\pi} e^{-2\pi i z}f(z). Homework Equations Nothing specific I can think of; general complex analysis/summation techniques. The Attempt at a Solution f(z+i) = \sum_{n...

I do believe we should consider the multiplicity; my language was imprecise as I was paraphrasing the problem I've encountered in several sources before. I think f(z)=z^n are precisely the types of functions we are looking for. But I have no clue how to show that any arbitrary analytic function...

Let f(x) be a function which is defined in the open unit disk (|z| < 1) and is analytic there. f(z) maps the unit disk onto itself k times, meaning |f(z)| < 1 for all |z| < 1 and every point in the unit disk has k preimages under f(z). Prove that f(z) must be a rational function. Furthermore...

Let A be an n*n matrix. Consider the space span \{ I, A, A^2, A^3, ... \} . How would one show that the dimension of the space never exceeds n? I feel like the answer lies somewhere near the Cayley-Hamilton theorem, but I can't quite grasp it.

Let A be a nonzero matrix of size n. Let a k*k submatrix of A be defined as a matrix obtained by deleting any n-k rows and n-k columns of A. Let m denote the largest integer such that some m*m submatrix has a nonzero determinant. Then rank(A) = k. Conversely suppose that rank(A) = m. There...

So c = (-λ_a_k)/(-λ_b_k), thanks!

Homework Statement 3y'' -y' + (x+1)y = 1 y(0) = y'(0) = 0 Homework Equations Not sure, that's the issue The Attempt at a Solution I can't quite get this one using the methods I'm familiar with, and I can't guess a particular solution to neither the equation nor the...

We can express any circle in the complex plane as |z-a|=k|z-b| where a and b are distinct complex numbers, k > 0 and k \not= 1. Is there an elegant way of showing this fundamental property of the complex plane to be true?

If A and B are both invertible square matrices of the same size with complex entries, there exists a complex scalar c such that A+cB is noninvertible. I know this to be true, but I can't prove it. I tried working with determinants, but a specific selection of c can only get rid of one entry...

I've found out through wolfram alpha that the inequality holds for an area enclosed by two crossing lines. Not quite sure where to get the two lines from

Homework Statement Determine the values of z \in \mathbb{C} for which |z+2| > 1 + |z-2| holds. Homework Equations Nothing complicated I can think of. The Attempt at a Solution For real values this holds for anything greater than 1/2. If I could figure out the boundaries of the...

It worked! Thanks, that was a really clever and elegant solution

Homework Statement \lim_{z \to 0} (\frac{sinz}{z})^{1/z^2} where z is complex Homework Equations The standard definition of a limit L'hopital's rule? The Attempt at a Solution I'm quite stumped by this one. There doesn't seem to be a way to break it down into different limits...

Actually, I am still running into some problems. Along the other straight side the x term can be expressed as x = re^{2\pi/n} where r is the distance from the origin. Clearly \frac{1}{1+x^{n}} = \frac{1}{1+r^{n}}, so integrating the side from 0 to R with respect to r will result in the same...