Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 13E (Further Complex Methods)

With $z=\cos x$,The chain rule givesHence $w_{xx}+n^{2}w=0$ becomes

A Papperitz symbol records the three regular singular points of a second-order Fuchsian equation and the two characteristic exponents at each point. Near $z=1$, substituting $w\sim (z-1{)}^{\rho }$ into the leading terms givesso the exponents are $0$ and $1/2$. The same calculation at $z=-1$ gives $0$ and $1/2$. At infinity, $w\sim z^s$ givesso $s=\pm n$. Since an exponent $\rho$ at infinity denotes behavior $z^{-\rho }$, these are recorded as $-n,n$. Thus the equation has P-symbolThe exponent sum is $1$, as required by the Fuchs relation for three singular points.

SetThis Möbius change sends $z=1,-1,\infty$ to $u=0,1,\infty$, respectively, without changing the corresponding exponents. The hypergeometric P-symbol has exponentsMatching it with $(0,1/2)$ at both finite singularities and $(-n,n)$ at infinity givesThe exponent-zero solution normalized to one at $u=0$ is therefore

The second standard local solution at $u=0$ isAfter absorbing the constant $2^{-1/2}$ into its normalization, this becomesTheir exponents at $z=1$ are $0$ and $1/2$, so they are linearly independent.

Take $n=1$ in part (c). The original equation $w_{xx}+w=0$ has the solution $\sin x=\sqrt{1-z^2}$. The solution $w_2$ is normalized by $w_2/(1-z{)}^{1/2}\to 1$ as $z\to 1$, soComparing this with the formula for $w_2$ and using $u=(1-z)/2$ givesThis also agrees with the general identity $F(A,C;C;u)=(1-u{)}^{-A}$.