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Cauchy-Riemann equations

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June 18th, 2023

Complex Differentiation

1) Suppose that an analytic function f defined on the whole of C satisfies Re(f(z))=0 for all z in C. Show that f is constant.

2i) Verify that u=x2-y2-y is harmonic in the whole complex plane.

2ii) Suppose f(x,y)=u(x,y)+iv(x,y). The Cauchy-Riemann equation state that: ux=vy and uy=-vx. For u=x2-y2-y compute ux and uy and use the Cauchy-Riemann equations to find v. The function v is called a conjugate harmonic function for u.

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