0
Who ever wrote this question, get a life and its not really a question!
Wiki User
∙ 14y ago
Add your answer:
Earn +20 pts
Show an analytical function of constant absolute value is constant?
This is based on exercise 10 on p 165.4 of "Introduction to Complex Analysis" by Nevanlinna and Paatero, Chelsea Publishing, NY. With w(z) == u(x,y) + i v(x,y), ( |w(z)| = constant ) ==> 1) uu + vv = const. Using notation ux == du/dx, uy == du/dy, etc., 1) ==> 2) 0 = u ux + v vx and 0 = u uy + v vy. Using the Cauchy-Riemann equations, ux = vy and uy = -vx to eliminate derivatives of v, 2) becomes 3) 0 = u ux - v uy and 0 = u uy + v ux. Exercise 10 asserts that from this one can show that 4) 0 = (uu + vv)(ux ux + uy uy). I have not obtained that formula, but using 3) one can form the combination 5) 0 = (u ux - v uy)(u ux + v uy) + (u uy + v ux)(u uy - v ux) which simlifies to 6) 0 = (uu - vv)(ux ux + uy uy). If w(z) is not a constant, then from the continuity of derivatives of analytic functions there must exist some domain throughout which ux or uy is non-zero. 6) indicates that in that domain either w(z) must vanish or its argument must be constant (45 or 225 degrees) so that in that domain w(z) must vanish or its modulus and argument must both be constant making w(z) constant. But if w(z) is constant in a domain, then by the uniqueness theorem (Nevanlinna and Paatero page 139) it is constant altogether.
Can somebody complete this I v uy your D I gm we ea to yj a ulm s o p c l e?
if you learned how to speak properly, people might be able to help you
What has the author Antolin V Uy written?
Antolin V. Uy has written various books on leadership, management, and personal development, including "The Leader Who Meets Employee Needs: A Practical Guide to Employee Engagement". Uy's works often focus on empowering leaders to build strong relationships with their employees and improve organizational effectiveness.
What are the initials for Uruguay?
UY
What is the birth name of Victoria Uy?
Victoria Uy's birth name is Victoria Bun Uy.
When was Rolando Uy born?
Rolando Uy was born in 1954.
How tall is Dustin Uy?
Dustin Uy is 5' 8".
How tall is Jocelyn Uy?
Jocelyn Uy is 5' 1".
How tall is Victoria Uy?
Victoria Uy is 5' 9".
When was Liz Uy born?
Liz Uy was born in 1982.
How tall is Benedikt Uy?
Benedikt Uy is 188 cm.
V O I D?
V O I D
