Suppose f1 and f2 are two analytic functions so that Re(f1)=u and Re(f2)=u. Then h(z)=f1(z)-f2(z) is analytic and by the Cauchy-Riemann equations (partial x Im(h), partial y Im(h))=0, which implies f1 and f2 only differ by a real constant. Can you explain this to me and show how this was arrived at? Thanks!

Suppose f1 and f2 are two analytic functions so that Re(f1)=u and Re(f2)=u. Then h(z)=f1(z)-f2(z) is analytic and by the Cauchy-Riemann equations (partial x Im(h), partial y Im(h))=0, which implies f1 and f2 only differ by a real constant. Can you explain this to me and show how this was arrived at? Thanks!

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Similar questions

Essentials of DISCRETE MATHEMATICS
6
Without using the Cauchy-Riemann equations, investigate the differentiability of the function f(z) = z Rez
Exercise 1- Determine which of the following functions are analytic: 2 2 d- x²+iy², b = 2xy + i(x²y²) 1 d= (2-1)(2+1) x-iy x²+y2 E- Sinx coshy+icosx sinhy C- )
Two waves on a string are given by the following functions: yı (x, 1) = 4 cos(201 – 30x) (cm) y2(x, 1) = -4 cos(201 + 30x) (cm) where x is in centimeters. The waves are said to interfere constructively when their superposition lys| = ly1 + y2|maximum, and they interfere destructively when lys| is a minimum. (a) What are the directions of propagation of waves y1 (x.f) and y2(x, t1)? (b) At t = (= (T/50) s, at what x location y do the two waves interfere constructively, and what is the corresponding value of ys|? (c) At t = (7/50) s, at what locationx do the two waves interfere destructively, and what is the corresponding value of lys?
In the function f(x, y) = ((2x + 3y)^10), we need to know the derivative in relation to the variable y. a) 30(2x - 3y)^9 b) 30(-2x+3y)^9 c) -30(2x+3y)^9 d) 30(2x+3y)^9 e) 30(-2x-3y)^9
Verify directly that the real and imaginary parts of the following analytic functions satisfy Laplace’s equation: (a) f(z) = z 2 + 2z + 1 (b) g(z) = 1 z (c) h(z) = e z
Compute the gradients of each of the following functions: o = xyz & = ln [(x² - y²)-¹/2] r = || = √√√x² + y² + x² 1/r Note: Remember that your tutorial has Hints in it, at the end of each chapter. If you're stuck, they are often quite useful! • If the answer is a scalar, you can just type it in the box (using the Calcpad if you like, or using / for fractions, ^ for exponents, shift and - for subscripts, etc.) For multiplication, you can either leave a space, or use *. So x*x=xx= x², Note that this is not the same as xx without a space; that will get read as an entirely different variable! • If you need to enter a vector, enter an ordered list of components, so for A you can enter either A = (A₂, Ay, A₂) = {A₂, Ay, A₂}. Note that the system isn't great with multiplying through by overall factors, so it's better not to write e.g. Vo= V = Vr = V= (2A, 2A, 2A₂).
Use the Cauchy-Riemann equations to show that
8. Which of the following could be differentiated by using implicit rule only? A) x2 + y2 = 9 B) 4x2y - xy = 3x C) 3x5y 50 D) 3xy – y? = 14 9. Which of the following need to contain when applying the implicit differentiation? A) x?y B) 5x C) 5* D) 2x-x2