3. Let u(x, y) = x² – y² and v(x, y) = x' -3xy. Show that u and v are harmonic functions but that their product uv is not harmonic. Show that u(x, y) = 2x-x+3xy is harmonic and find a harmonic conjugate 4.
3. Let u(x, y) = x² – y² and v(x, y) = x' -3xy. Show that u and v are harmonic functions but that their product uv is not harmonic. Show that u(x, y) = 2x-x+3xy is harmonic and find a harmonic conjugate 4.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Question 3
Question 4
![1.
Let a functionf be analytic everywhere in a domain D. Suppose that f(z) is pure
imaginary for all z in D. What can we conclude about the values of f(z)?
(Hint: Use the theorem or the first corollary presented in Lecture 10)
Let f and g be analytic functions in a domain D. If f'(z) = g'(z) for all z in D,
then show that f(z)= g(z)+c, where c is a complex constant.
2.
3.
Let u(x, y) = x2 - y² and v(x, y) = x' - 3xy². Show that u and v are harmonic
functions but that their product uv is not harmonic.
Show that u(x, y) = 2x-x'+3xy is harmonic and find a harmonic conjugate
v(x, y).
4.
Show that exp(z) s exp(=|") for all z E C.
5.
6.
Show that Log[(-1+i)*]#2Log(-1+i).
7.
Find all roots of the equation log(z) = ni / 2
8.
Find the principal value of (1+ i)' .
9.
Use the definitions of sin(z) and cos(z) given in Lecture 13 at the 13:30 mark to
cos(z) for all z e C.
%3D
prove that Sin z+
(Do not use any other trigonometry identities for question 9)
1
10.
Evaluate (3t-i)dt](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcfc5bdab-6654-444e-b9d5-5fc4b8470fbb%2F6f7650a0-015c-409c-91fb-3548fdc4b53d%2Fnd7xvrr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1.
Let a functionf be analytic everywhere in a domain D. Suppose that f(z) is pure
imaginary for all z in D. What can we conclude about the values of f(z)?
(Hint: Use the theorem or the first corollary presented in Lecture 10)
Let f and g be analytic functions in a domain D. If f'(z) = g'(z) for all z in D,
then show that f(z)= g(z)+c, where c is a complex constant.
2.
3.
Let u(x, y) = x2 - y² and v(x, y) = x' - 3xy². Show that u and v are harmonic
functions but that their product uv is not harmonic.
Show that u(x, y) = 2x-x'+3xy is harmonic and find a harmonic conjugate
v(x, y).
4.
Show that exp(z) s exp(=|") for all z E C.
5.
6.
Show that Log[(-1+i)*]#2Log(-1+i).
7.
Find all roots of the equation log(z) = ni / 2
8.
Find the principal value of (1+ i)' .
9.
Use the definitions of sin(z) and cos(z) given in Lecture 13 at the 13:30 mark to
cos(z) for all z e C.
%3D
prove that Sin z+
(Do not use any other trigonometry identities for question 9)
1
10.
Evaluate (3t-i)dt
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