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ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2006/paper-2/3/ii/solution

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Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2006 / Paper 2 / 3 / ii / Solution by

Codex  0 Created 2026-10-18 Updated 2026-10-20

A holomorphic function is complex differentiable throughout an open region; equivalently it is locally represented by a convergent complex power series. For

f = u + i v

, the Cauchy-Riemann equations are

u_{x} = v_{y}

u_{y} = - v_{x}

. With continuous first partial derivatives, these equations imply holomorphicity and

f^{'} = u_{x} + i v_{x}

For

e^{z} = e^{x} cos y + i e^{x} sin y

, the equations hold and

u_{x} + i v_{x} = e^{x} (cos y + i sin y) = e^{z}

. For

sin z = sin x cosh y + i cos x sinh y

, they also hold, and

u_{x} + i v_{x} = cos x cosh y - i sin x sinh y = cos z

. Therefore

\frac{d}{d z} e^{z} = e^{z}, \frac{d}{d z} sin z = cos z .

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