 Home

Codex Wiki

List of universities

University of Cambridge

Course of the University of Cambridge

Natural sciences course of the University of Cambridge

Year of the natural sciences course of the University of Cambridge

Year 2 of the natural sciences course of the University of Cambridge

Mathematics course of the year 2 of the natural sciences course of the University of Cambridge

Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge

2025

Paper 1

The <Cauchy-Riemann equations> determine the <harmonic conjugate>. First
$$
u_x=e^x[(x+1)\cos y-y\sin y],
$$
so $v_y=u_x$ integrates with respect to $y$ to
$$
v=e^x(x\sin y+y\cos y)+h(x).
$$
Now
$$
u_y=e^x[-(x+1)\sin y-y\cos y],
$$
and the second <Cauchy-Riemann equation>, $v_x=-u_y$, forces $h'(x)=0$. Hence
$$
f=e^x\{x\cos y-y\sin y+i(x\sin y+y\cos y)\}+iC=ze^z+iC.
$$
The normalization at zero gives $C=0$, so the required <entire function> is
$$
\boxed{f(z)=ze^z.}
$$

Solution

Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2025 / Paper 1 / 7 / b / Solution

Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2025 / Paper 1 / 7 / b / Solution

 Ancestors (13)

 Discussion (0)

 Articles by others on the same topic (0)

Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2025 / Paper 1 / 7 / b / Solution

 Ancestors (13)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)