Question

How to prove a function is bounded or not in complex analysis?

Answer 1

Using Cauchy's integral formula, write

f(n)(a)=12πi∫|z|=Rf(z)(z−a)n+1dz

Apply absolute value on each side to get:

|f(n)(a)|=∣∣∣12πi∫|z|=Rf(z)(z−a)n+1dz∣∣∣≤12π∫2π0∣∣∣f(Reit)(Reit−a)n+1iReit∣∣∣dt=R2π∫2π0|f(Reit)||Reit−a|n+1dt≤R2π∫2π0|Reit|−−−−−√(R−|a|)n+1dt=R3/2(R−|a|)n+1

Now observe that for all n≥1, limR→∞R3/2(R−|a|)n+1=0.

From here you can easily conclude the proof.

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