Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
(a) Define the symbols ôf/DE and af/dz by af 1 (af 10f 2 (dr af 1 (of 18f i by, dz 2 dr' i dy as suggested by the relations z (:+ =), y = ±(: - =) and the chain rule. Show that the Cauchy-Riemann equations are equivalent to Of/0 = 0. Also, show that if f is analytic, then f'= 0f/0z.

(a) Define the symbols ôf/DE and af/dz by af 1 (af 10f 2 (dr af 1 (of 18f i by, dz 2 dr' i dy as suggested by the relations z (:+ =), y = ±(: - =) and the chain rule. Show that the Cauchy-Riemann equations are equivalent to Of/0 = 0. Also, show that if f is analytic, then f'= 0f/0z.

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
See similar textbooks
icon
Related questions
Question
2. (Cauchy-Riemann's equations, analyticity and harmonic functions) (a) Define the symbols ðf /dž and ôf /dz by af_1 (af 1af dz 2 dx i dy, af _1(af ¸ 1af dz 2 (dr" i dy, as suggested by the relations z = }(2+ 2), y = ±(z – =) and the chain rule. Show that the Cauchy-Riemann equations are equivalent to df/ðz = 0. Also, show that if f is analytic, then f' = df/dz. (b) Determine all functions f = u + iv that are analytic in the whole plane and has the property that the real part u is a function of only y = Im z. The answer should be given as an expression in the variable z = x + iy.
Transcribed Image Text:2. (Cauchy-Riemann's equations, analyticity and harmonic functions) (a) Define the symbols ðf /dž and ôf /dz by af_1 (af 1af dz 2 dx i dy, af _1(af ¸ 1af dz 2 (dr" i dy, as suggested by the relations z = }(2+ 2), y = ±(z – =) and the chain rule. Show that the Cauchy-Riemann equations are equivalent to df/ðz = 0. Also, show that if f is analytic, then f' = df/dz. (b) Determine all functions f = u + iv that are analytic in the whole plane and has the property that the real part u is a function of only y = Im z. The answer should be given as an expression in the variable z = x + iy.
Expert Solution
steps

Step by step

Solved in 4 steps

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Similar questions
  • SEE MORE QUESTIONS
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS