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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2018

Paper 2

On the cut plane choose $0<\theta<2\pi$. For $z=x+iy$,
$$
u(x,y)=\frac12\log(x^2+y^2),
\qquad v(x,y)=\arg(x+iy).
$$
Their derivatives are
$$
u_x=\frac{x}{x^2+y^2}=v_y,
\qquad
u_y=\frac{y}{x^2+y^2}=-v_x.
$$
The <Cauchy-Riemann equations> hold, so this branch is holomorphic. Its derivative is
$$
\boxed{\frac{dw}{dz}=u_x+iv_x
=\frac{x-iy}{x^2+y^2}=\frac1z.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 2 / 13B / b / iii / Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 2 / 13B / b / iii / Solution

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Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 2 / 13B / b / iii / Solution

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