Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 13D (Further Complex Methods)

An elliptic function is a meromorphic $f:\mathbb{C}\to \hat{\mathbb{C}}$ with two real-linearly independent periods ${\omega }_1,{\omega }_2$. A fundamental cell may be taken as the half-open parallelogramwith $z_0$ chosen so that its boundary contains no zero or pole under discussion. The argument principle and cancellation on opposite sides show that a nonconstant elliptic function has the same number of zeros and poles in a cell, counting multiplicity.

If an elliptic function has no poles, it is entire. It is bounded on the closure of a fundamental parallelogram, and periodicity transfers that bound to the whole plane. Liouville's theorem therefore makes it constant.

Choose a fundamental parallelogram with no pole on its boundary. Opposite sides are translates traversed in opposite directions, so periodicity givesThe residue theorem then givesA lone simple pole has a nonzero residue by definition of its order, so an elliptic function cannot have exactly one simple pole in a fundamental cell.

Fix $w\in \mathbb{C}$ and apply the stated clockwise argument principle to $h=f-w$. Its logarithmic derivative is periodic, so the integrals over opposite sides of a fundamental cell cancel. Hence $P-Z=0$. The poles of $f-w$ are precisely the poles of $f$, with unchanged multiplicities, while its zeros are the solutions of $f(z)=w$. Thus the value multiplicity of an elliptic function is the same fixed pole count for every finite $w$, counting multiplicity.

At every lattice point $\omega =m{\omega }_1+n{\omega }_2$, the translated summand supplies a principal partAll other terms are holomorphic nearby. Hence the Weierstrass elliptic function has a double pole at every lattice point, with zero residue, and no other singularities.

For a nonzero lattice point $\omega$ and $|z|<|\omega |$,Pairing $\omega$ with $-\omega$ cancels odd powers. There is no constant term, so $a_0=0$, and the laurent coefficients of the Weierstrass elliptic function are