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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 信 du dv %3D de ne 18 ||

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 信 du dv %3D de ne 18 ||

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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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
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.
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