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Pick out the true statements:

(a) There exists an analytic function $f$ on $\mathbb{C}$ such that $f(2i) = 0$, $f(0) = 2i$ and $|f(z)|\le 2$ for all $z\in\mathbb{C}$

. (b) There exists an analytic function $f$ in the open unit disc $\{z\in\mathbb{C} : |z| < 1\}$ such that $f(1/2) = 1$ and $f(1/2^n ) = 0$ for all integers $n\ge 2 $.

(c) There exists an analytic function whose real part is given by $u(x, y) = x^2 + y^2$, where $z = x + iy$.

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clearly c is not true . what about a and b.c is not true as v cannot be find using cauchy riemann equation –  poton Aug 9 '12 at 17:40

2 Answers 2

up vote 3 down vote accepted

a) by Liovilles theorem $f$ is constant

b) by Identity theorem of Analytic function, $f\equiv 0$

c) you have done it.

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Yes @poton is right, real part should be harmonic. And (a) is not true either because the Liouville theorem (bounded analytic functions are constant), as for (b), the isolated zeros theorem insures that $f$ equals to zero so (b) is false also ...

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