- #1

claralou_

- 5

- 0

## Homework Statement

For R > 0,

assume ΓR is a circle {z ∈ C : |z| = R} with anticlockwise direction.

For which R>0, does the the function f(z) = 1/sin^(2)(z) be continuous on ΓR

and evaluate ∫_{ΓR} dz/sin^(2)(z) for each R (the answer may be dependent on R).

## Homework Equations

sinx= (e^(ix) - e^(-ix)) / 2i (possibly)

Resf(z)= lim (pole x f(z))

## The Attempt at a Solution

used 1/e^(iz) = sin(x)

so found

1/sin^{2}x = 1 / (e^{2iz}

Began using the

closed integral over C of f(z)dz = Integral from -R to +R of f(x)dx + integral f(z) dz

Found that there was a pole at z = 1/2i and found the residue at that point to be 1/(2ie)