Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 3 / 8B (Further Complex Methods)

Let $u_1,u_2$ be a local fundamental pair at zero, including a logarithmic Frobenius solution if the exponents are resonant. Any other independent pair on a common branch domain has $w_1=a{u}_1+b{u}_2$ and $w_2=c{u}_1+d{u}_2$ with constant coefficients and $ad-bc\ne 0$. Bilinearity and antisymmetry of the Wronskian giveThus changing the independent branches multiplies the principal-pair Wronskian by a nonzero constant, once the branches are continued onto the same domain.

For the second-order Fuchsian equation represented by the Papperitz symbol, the coefficient of $w^{\prime }$ isThis follows from the sums of local exponents at its two finite regular singular points. Abel's Wronskian identity, obtained by differentiating $w_1{w}_2^{\prime }-w_1^{\prime }{w}_2$ and using the differential equation, gives $W^{\prime }=-pW$. Thus, on every branch domain avoiding zero and one,The normalized expression is locally constant. Its analytic continuation around either singularity is unchanged: the power factors cancel the determinant of the local monodromy, equivalently the integral of $-p$ around that singularity. It therefore defines a single-valued holomorphic function on the punctured plane, locally constant near both punctures. Those punctures are removable, so the normalized expression extends to an entire function. This argument also covers resonant exponents and avoids assuming two pure power solutions in that case.

Let $H=z^{1-\alpha -{\alpha }^{\prime }}(1-z{)}^{1-\gamma -{\gamma }^{\prime }}W$. The exponent-sum relation gives $2-\alpha -{\alpha }^{\prime }-\gamma -{\gamma }^{\prime }=1+\beta +{\beta }^{\prime }$. Therefore the assumed boundedness of $z^{1+\beta +{\beta }^{\prime }}W$ at infinity makes $H$ bounded there, up to a fixed nonzero branch factor. By part (ii), $H$ is entire and hence bounded also on compact sets. Liouville theorem makes it a constant $A$, givingIndependence forces $A\ne 0$: if the Wronskian vanished, uniqueness for the second-order linear equation at an ordinary point would make the two solutions dependent. In fact the differential identity in part (ii) already proves constancy; the boundedness hypothesis supplies the requested Liouville route as well.