Let \( f \) be a holomorphic function on the disc \( D_{R_0} \) centered at the origin and of radius \( R_0 \).(a) Prove that whenever \( 0 < R < R_0 \) and \( |z| < R \), then\[f(z) = \frac{1}{2\pi} \int_{0}^{2\pi} f(Re^{i\varphi}) \operatorname{Re} \left( \frac{Re^{i\varphi} + z}{Re^{i\varphi} - z} \right) d\varphi.\]There are no graphs or diagrams accompanying this text.

Here we have given that f be a holomorphic function on the disc DR0 centered at the origin and of…

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Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Let f be a holomorphic function on the disc DR centered at the origin and of radius Ro. (a) Prove that whenever 0 < R< Ro and |z| < R, then 2T 1/2 for f(Re Re (Rebe 2π f(z) = Re+z) +2) dip. Reipz.