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Using Ex 9.5, is cos|Z|^4 and Im Z entire? Why?

Using Ex 9.5, is cos|Z|^4 and Im Z entire? Why?

Advanced Engineering Mathematics
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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Using Ex 9.5, is cos|Z|^4 and Im Z entire? Why?

Ex 9.5: Prove: if f(z) is analytic and real on a domain ✓ then I is a constant function. Pf: & is analytic and real on √✓ => f(x,y) = u(x, y) + i.o namely V(x, y) = 0; CR equations hold for every Z EN I from analyticity OV on √r}: = 0 and 04-0 ry Ox oy 24 = Du ди ~/y = 0 => H(X,Y) = CER =) f(x, y) = c, namely f is a constant function.
Transcribed Image Text:Ex 9.5: Prove: if f(z) is analytic and real on a domain ✓ then I is a constant function. Pf: & is analytic and real on √✓ => f(x,y) = u(x, y) + i.o namely V(x, y) = 0; CR equations hold for every Z EN I from analyticity OV on √r}: = 0 and 04-0 ry Ox oy 24 = Du ди ~/y = 0 => H(X,Y) = CER =) f(x, y) = c, namely f is a constant function.
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