Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Suppose that f E H(C) and |f(z)| < ekez for all z. Show that f(z) =
ce? for some constant c.
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Transcribed Image Text:Suppose that f E H(C) and |f(z)| < ekez for all z. Show that f(z) = ce? for some constant c.
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Solution:

Consider fz=cez for z=x+i y

Take absolute value on both sides:

fz=cez=cex+iy=cexeiy=cex         eiy=1

We know, in polar form x=rcosθ,

fz=cercosθ=cerecosθ=ecosθ if c=1ereRez

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