Recent content by StumpedPupil

I'm sorry, that is a typo. I meant to write Re((1 + i)w) = 0. This is the line y = x. I choose two points on this line, and a point at infinity.

Homework Statement Define L: |z| = 1 -----> Re( (1 + w)) = 0. Find L. Homework Equations A transformation is defined by three unique points by T(z) = (z-z1)(z2-z3) / (z-z3)(z2-z1). If we have two transformations T and S, and we want T = S for three distinct points, then we have the...

Ok, yeah I foolishly wrote the answer is arcsine but it is actually π/4 * 1/sin(3π/8).

Homework Statement Calculate the integral [ z^4/(1 + z^8) ] over negative infinity to positive infinity. Homework Equations Residue Theorem. Specifically for real-valued rational functions (on the real axis) where the denominator exceeds the degree of the numerator by at least two or...

Yes, only four poles are in the upper half plane, namely the values for the variable w, which I mistakenly labeled as a z when z already existed. So I will integrate over the semi circle for very large R in the upper half plane for the poles z = e^(πiw), where w =1/8, 3/8, 5/8 and 7/8. Thank...

Homework Statement the integral from negative infinity to positive infinity: z^4/(1 + z^8)dz Homework Equations The residue theorem: <http://en.wikipedia.org/wiki/Residue_theorem>. The Attempt at a Solution I found the 8th roots of z^8 = -1, which are e^(πiz), where z =1/8, 3/8...

It definitely helps, thank you. I do have a follow up question though. In the general case where F has a zero of order m and we represent F = (z-z0)^mg(z), like you did, then we can get F'/F = m/(z-z0) + g'(z)/g(z). The only condition of g(z) is that it is analytic, so is it not possible...

Homework Statement Show Res(F'(z)/F(z), z0) = m if F(z) is analytic on the disc |z - z0| < R and has a zero of order m at z0. Homework Equations The Attempt at a Solution We know that the kth derivation of F(z) is 0 for all k less than m, since F(z) has a zero of order m at z0...

Homework Statement How does Cauchy's Formula help find the power series of the complex function f(z) = e^z. Homework Equations e^z = ∑z^k/k! (sum from k = 0 to infinity) Cauchy's Formula Consequence of Cauchy's Formula: F(z) is analytic in a domain D and the point z1 is in D. If...

That doesn't actually help because you end with the same results. The denominator becomes (3 - cos(2θ)), which you can derive ∫4iz/(z^4 - 6z^2 + 1)dz from. Thanks for the suggestion though.

Homework Statement ∫dθ/(1 + (sinθ)^2 ), [0, π] Homework Equations Cauchy's Formula, perhaps Cauchy's Thm. http://mathworld.wolfram.com/CauchyIntegralFormula.html http://mathworld.wolfram.com/CauchyIntegralTheorem.html The Attempt at a Solution I first substituted sinθ =...