# Maths Homework #3

Problem 1Contour C is a circle of radius 3 centered at the origin, f(z) = coszz(z?1). Compute HCf(z) dzby two methods.(a) Decompose 1z(z?1) into partial fractions and apply Cauchyâs integral formula twice.(b) Do not split into partial fractions. Instead, deform the contour so that integrationof f(z) is done over two small circles, one centered at the origin, and the other centeredat z = 1. Do you get the same result? Draw a picture and provide a short explanationof how exactly the contour deformation works in this case and why it yields the correctvalue.Problem 2Using the same method we used in class to compute R???ei?x1+x2 dx, compute the followingintegral for any ? ? 0:Z???ei?x(x + 1 + i)(x ? 1 + i)dx = …Hint: Decomposing into partial fractions may help.Problem 3Contour C is a circle of radius 100 centered at 10 + 2i. Expand function f(z) = z cosz(z??)3into Laurent series about ? and evaluateICz cos z(z ? ?)3dz.Hint: z = (z ? ?) + ?, cos(z ? ?) = …Problem 4(a) Expand function f(z) = sin((z?3)2)(z?4) into the Taylor series about 3; ?nd ?rst 3 non-zeroterms AND determine the radius of convergence for this series. Hint: z ? 4 = (z ? 3) ? 1.(b) Expand function f(z) = ez(z+1)(z?2)2into the Laurent series about 2. Find ?ve terms (theones with the lower powers of (z ? 2)).