HAND-IN SET 2, FYS3140 spring 2020 DUE: 31/1

NB! The problem set consists of two pages

Problem H2.1 (Cauchy-Riemann/Harmonic functions)

Let v(x, y) = x2 + 2xy − y2 be the imaginary part of an analytic function f(z). Moreoverwe are given that f(0)=0. Find the real part and write f(z) compactly.

Problem H2.2 (Cauchy-Riemann conditions)

Find the Cauchy-Riemann conditions in polar coordinates.Hint: Write z = reiθ and f = u(r, θ) + iv(r, θ). Like for cartesian coordinates, the deriva-tion is based on demanding that the derivative df/dz has to be unique, independent of ∆z.In this case, let ∆z be along the radial and tangential direction, respectively.

Problem H2.3 (Cauchy theorem and ”important integral”)

Evaluate the integral

I =

∮Γ

2z2 − z + 1

(z − 1)2(z + 1)dz (1)

where Γ is the contour shown in the figure.Hint: Use partial fraction decomposition (”delbrøksoppspalting”). Deform the contourappropriately.

Figure 1: The contour for problem H2.3: A figure-eight contour traversed once in thedirection indicated.

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HAND-IN SET 2, FYS3140 spring 2020 DUE: 31/1

Problem H2.4 (Cauchy theorem and integral formula)

Compute the integral

I =

∮Γ

z + i

z3 + 2z2dz (2)

where the contour Γ is

a) the circle |z + 2− i| = 2 traversed once, counterclockwise

b) the circle |z| = 1 traversed once, counterclockwise

c) the circle |z − 2i| = 1 traversed once, counterclockwise

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