Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2023 / Paper 1 / 7

For an analytic function, the complex differentiability limit is independent of the direction of approach. A real increment gives $f^{\prime }(z)=u_x+i{v}_x$, whereas an imaginary increment givesEquality of real and imaginary parts proves the Cauchy-Riemann equations:Conversely, if $u,v$ have continuous first partial derivatives near a point and obey these equations, their real differentiability gives, for a complex increment $h$,Dividing by $h$ proves the complex differentiability there. Continuous first partial derivatives are a sufficient regularity hypothesis for this converse; agreement along the two coordinate directions alone is only a necessary condition.

Using the complex cosine identity,we findIts real part is exactly the prescribed function. Thus a harmonic conjugate is $v=y\cos x\cosh y-x\sin x\sinh y$, and the general answer isTo check the only freedom, two analytic functions with this real part differ by a function with zero real part. The Cauchy-Riemann equations then force both partial derivatives of its imaginary part to vanish, so it is constant on the connected domain. No normalization was supplied to fix $C$.

Multiply the exponential power series by the two prefactors:Collecting powers yields the Laurent seriesIts first terms are $z+1+3/(2z)+7/(6z^2)+13/(24z^3)+\cdots$. Infinitely many negative-power coefficients are nonzero, so $\boxed{z=0\text{ is an essential singularity}}$. The residue is the coefficient of $z^{-1}$:The phrase "power series" here refers to a Laurent series; no ordinary Taylor series about an essential singularity exists.

The denominator vanishes at $z=\pm i\sqrt{2}$, giving two simple poles. Their distances from the center are both $|1\mp i\sqrt{2}|=\sqrt{3}$. There are no nearer singularities, so the Taylor series has radius of convergence

Apply the Cauchy-Hadamard theorem to the power series coefficients $a_n=[n/(n+\alpha ){]}^{n^2}$ for $n\ge 1$:For example, taking logarithms gives $-n\log (1+\alpha /n)\to -\alpha$. ThereforeAny convention for the isolated $n=0$ coefficient has no effect on the radius of convergence.