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Suppose that f = u+ iv is analytic in a domain D. Show that the Cauchy-Riemann equations in polar coordinates (r, 0), where I = r cos 0, y =r sin 0, are given by dv ar ne

Suppose that f = u+ iv is analytic in a domain D. Show that the Cauchy-Riemann equations in polar coordinates (r, 0), where I = r cos 0, y =r sin 0, are given by dv ar ne

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.
5. Suppose that f = u+ iv is analytic in a domain D. Show that the Cauchy-Riemann equations in polar coordinates (r, 0), where x = r cos 0, y = r sin 0, are given by du dv ne ar
Transcribed Image Text:5. Suppose that f = u+ iv is analytic in a domain D. Show that the Cauchy-Riemann equations in polar coordinates (r, 0), where x = r cos 0, y = r sin 0, are given by du dv ne ar
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