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1. Suppose that f is analytic on a domain D and Vz € D,f'(z) = af(z) for some constant a. Show that f(z) = cexp(az) for some constant c E C. (Hint: you might want to First, show that if g(z) is analytic on a domain D with g'(z) = 0 for all z e D, then g is constant on D; • Next, consider g(z) = e¬a² f(2).

1. Suppose that f is analytic on a domain D and Vz € D,f'(z) = af(z) for some constant a. Show that f(z) = cexp(az) for some constant c E C. (Hint: you might want to First, show that if g(z) is analytic on a domain D with g'(z) = 0 for all z e D, then g is constant on D; • Next, consider g(z) = e¬a² f(2).

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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Analytic functions and their close relatives, harmonic functions, are the stuff of which the subject of complex variables is built. This section introduces both of these types of functions; here and in subsequent sections, many of their significant prop- erties are developed. A function f defined for z in a domain D is differentiable at a point zo in D if f(z) – f(zo) = lim f(zo + h) – f(zo) lim (1) z – Zo h z+zo h-0 exists; the limit, if it exists, is denoted by f'(zo). If f is differentiable at each point of the domain D, then f is called analytic in D. A function analytic on the whole complex plane is called entire.
Transcribed Image Text:Analytic functions and their close relatives, harmonic functions, are the stuff of which the subject of complex variables is built. This section introduces both of these types of functions; here and in subsequent sections, many of their significant prop- erties are developed. A function f defined for z in a domain D is differentiable at a point zo in D if f(z) – f(zo) = lim f(zo + h) – f(zo) lim (1) z – Zo h z+zo h-0 exists; the limit, if it exists, is denoted by f'(zo). If f is differentiable at each point of the domain D, then f is called analytic in D. A function analytic on the whole complex plane is called entire.
Suppose that f is analytic on a domain D and Vz e D,f'(2) = af(z) for some constant a. Show that f(2) = cexp(az) for some constant c € C. 4. (Hint: you might want to • First, show that if g(z) is analytic on a domain D with g'(z) = 0 for all z € D, then g is constant on D; • Next, consider g(z) = e-«² f(z).
Transcribed Image Text:Suppose that f is analytic on a domain D and Vz e D,f'(2) = af(z) for some constant a. Show that f(2) = cexp(az) for some constant c € C. 4. (Hint: you might want to • First, show that if g(z) is analytic on a domain D with g'(z) = 0 for all z € D, then g is constant on D; • Next, consider g(z) = e-«² f(z).
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