3.II.17E3 . \mathrm{II} . 17 \mathrm{E} \quad

Further Analysis | Part IB, 2004

(i) Explain why the formula

f(z)=n=1(zn)2f(z)=\sum_{n=-\infty}^{\infty} \frac{1}{(z-n)^{2}}

defines a function that is analytic on the domain C\Z\mathbb{C} \backslash \mathbb{Z}. [You need not give full details, but should indicate what results are used.]

Show also that f(z+1)=f(z)f(z+1)=f(z) for every zz such that f(z)f(z) is defined.

(ii) Write logz\log z for logr+iθ\log r+i \theta whenever z=reiθz=r e^{i \theta} with r>0r>0 and π<θπ-\pi<\theta \leqslant \pi. Let gg be defined by the formula

g(z)=f(12πilogz)g(z)=f\left(\frac{1}{2 \pi i} \log z\right)

Prove that gg is analytic on C\{0,1}\mathbb{C} \backslash\{0,1\}.

[Hint: What would be the effect of redefining logz\log z to be logr+iθ\log r+i \theta when z=reiθz=r e^{i \theta}, r>0r>0 and 0θ<2π0 \leqslant \theta<2 \pi ?]

(iii) Determine the nature of the singularity of gg at z=1z=1.

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