Recent content by bmanbs2

I didn't, but I just recieven an update to the assignment that says I need to show the matrices are normal.

Bump?

OK I understand what needs to be done and found the answer, so thank you so much. For posterity's sake however, I considered the term, \frac{1}{(z-2\pi)\left[\frac{1}{2!}-\frac{(z-2\pi)^2}{4!}+\cdots\right]}, Found the closed-form of the series, found its reciprocal, and turned it back into a...

OK I understand now, but is there an easier way to find that one coefficient other than lots and lots of algebraic manipulation?

All I know is that f(z) is analytic.

So does d/dz(Im(f(z))) = Im(f'(z))? For context, if Im(f'(z)) > 0, is Im(f(z)) always increasing and one-to-one? f(z) is analytic

That's not really helping. I can calculate [PLAIN]http://www.sharpermath.com/cgi/mimetex.cgi?\frac{1+z}{1-cos(z)}%20=%20\frac{1+z}{(z-2pi)[{\frac{(z-2\pi)}{2!}-\frac{(z-2\pi)^3}{4!}+\frac{(z-2\pi)^5}{6!}-+...]} But I guess I don't see how this helps. Sorry we've never used Laurent...

Homework Statement Solve the contour integral \int_{\gamma}\frac{2+z}{1-cos\left(z\right)}dz along a counter-clockwise circle of radius 1 centered at z=6 Homework Equations Cauchy's integral formula f\left(z\right)=\frac{1}{i*2\pi}\int_{\gamma}\frac{f\left(w\right)}{w-z}dw The...

Homework Statement Matrices A and B are simultaneously unitarily diagonalizeable. Prove that they commute. Homework Equations As A and B are simultaneously unitarily diagonalizeable, there exists a unitary matrix P such that P^{-1}AP = D_{1} and P^{-1}BP = D_{2}, where D_{1} and D_{2}...

Homework Statement Find the Linear Fractional Transformation that maps the line Re\left(z\right) = \frac{1}{2} to the circle |w-4i| = 4. Homework Equations For a transform L\left(z\right), T\left(z\right)=\frac{z-z_{1}}{z-z_{3}}\frac{z_{2}-z_{3}}{z_{2}-z_{1}}...

Actually just found the answer, so I'll post it here. Consider the function h = g - f. As f and g are analytic, h is also analytic. But as Re\left(f\right) = \left(g\right) on B, Re\left(h\right) = 0 on B. The Maximum Modulus Principle states that |Re\left(h\right)| reaches its maximum on...

Homework Statement Suppose f and g are analytic on a bounded domain D and continuous on the domain's boundary B. Also, Re\left(f\right) = Re\left(g\right) on B. Show that f = g + ia, where a is a real number. Homework Equations The maximum modulus principle states that Re\left(f\right) and...

Thanks for the reply, but I want to make sure I'm setting up the equations properly. So far I have that im\left(A_{i-1}\right) + im\left(A_{i}\right) = d_{i} im\left(A_{i}\right) + im\left(A_{i+1}\right) = d_{i+1} Subtracting the system of equations gives im\left(A_{i-1}\right) -...

Homework Statement Consider integer sequence n_{1},...,n_{r} and matrices A_{1},...,A_{n-1}. Assume im\left(A_{i}\right) = ker\left(A_{i+1}\right) Using the rank-nullity theorem, show that \sum^{n}_{i=1}\left(-1\right)^{i}d_{i} = 0 Homework Equations The rank-nullity theorem...

Reviving this tread, but why does \lim_{R\rightarrow \infty} \int_0^{\frac{\pi}{4}} e^{i R^2 e^{i 2 \theta}} i R e^{i\theta} d \theta = 0 ?