• Support PF! Buy your school textbooks, materials and every day products Here!

Cauchy-Riemann condition

  • Thread starter chwala
  • Start date
  • #1
chwala
Gold Member
618
42

Homework Statement


Verify ## f(z)= z^3-5iz+√7## satisfies cauchy riemann equations.


Homework Equations




The Attempt at a Solution


seeking alternative method
## f(z)= (x^3+5y-3xy^2+√7)+ (3x^2y-y^3-5x)i##
##∂u/∂x = 3x^2-3y^2 = ∂v/∂y##
##∂v/∂x=6xy-5= -∂u/dy##
hence satisfies.
 
Last edited by a moderator:

Answers and Replies

  • #2
jambaugh
Science Advisor
Insights Author
Gold Member
2,211
245
Your attempt appears correct. What was your question?
 
  • #3
33,271
4,976

Homework Statement


Verify ## f(z)= z^3-5iz+√7## satisfies cauchy riemann equations.


Homework Equations




The Attempt at a Solution


seeking alternative method
Why? Is there some reason that the work below isn't satisfactory?
chwala said:
## f(z)= (x^3+5y-3xy^2+√7)+ (3x^2y-y^3-5x)i##
##∂u/∂x = 3x^2-3y^2 = ∂v/∂y##
##∂v/∂x=6xy-5= -∂u/dy##
hence satisfies.
 
  • #4
Dick
Science Advisor
Homework Helper
26,258
618
seeking alternative method
If you think about expressing ##f=u+iv## in terms of ##z## and ##\bar{z}## instead of ##x## and ##y##, it's easy to show that the CR equations say that ##\frac{\partial{f}}{\partial{\bar{z}}}=0##. So any function that depends only on ##z## will satisfy CR.
 
  • #5
FactChecker
Science Advisor
Gold Member
5,384
1,953
There is a simple way to see it immediately if you know that the sums and products of functions that satisfy CR also satisfy CR. Constants and f(z)=z satisfy CR. Your equation is a combination of sums and products satisfying CR, so it also satisfies CR.
 
  • #6
chwala
Gold Member
618
42
Why? Is there some reason that the work below isn't satisfactory?
why? i have shown that both sides of the cauchy riemann pde hold. thats a summary of my working, or you want to see the full working?
 
  • #7
chwala
Gold Member
618
42
There is a simple way to see it immediately if you know that the sums and products of functions that satisfy CR also satisfy CR. Constants and f(z)=z satisfy CR. Your equation is a combination of sums and products satisfying CR, so it also satisfies CR.
thats exactly what i did. i substituted for ##z=x+iy##
 
  • #8
chwala
Gold Member
618
42
Your attempt appears correct. What was your question?
My method is correct, its not an attempt. I was proving or rather to show that the given ##f(z)## satisfies the Cauchy Riemann equations. I am simply asking for alternative ways of solving. regards,
 
  • #9
jambaugh
Science Advisor
Insights Author
Gold Member
2,211
245
Well as the Cauchy Riemann conditions are directly verifiable there's not much else you can do but directly verify them. There are consequential conditions like the harmonic condition:
[tex]\Delta u = \Delta v = 0[/tex]
but while this is a necessary condition it is not sufficient. It may be satisfied for functions failing to satisfy CR.
[edit: oops for the typo... now corrected.]
 
Last edited:
  • #10
chwala
Gold Member
618
42
Well as the Cauchy Riemann conditions are directly verifiable there's not much else you can do but directly verify them. There are consequential conditions like the harmonic condition:
[tex]\Delt u = \Delta v = 0[/tex]
but while this is a necessary condition it is not sufficient. It may be satisfied for functions failing to satisfy CR.
.....directly verifiable.........thank you.
 
  • #11
FactChecker
Science Advisor
Gold Member
5,384
1,953
thats exactly what i did. i substituted for ##z=x+iy##
What you did is not the same as applying general principles. You don't have to do the messy calculations if the principles have been proven. You have to have proven the principle that sums and multiplications of functions satisfying CR will also satisfy CR. Then just state that your function is a combination of sums and multiplications of simple functions satisfying CR and apply the principle.
 
  • #12
chwala
Gold Member
618
42
why? i have shown that both sides of the cauchy riemann pde hold. thats a summary of my working, or you want to see the full working?
kindly note pde means partial differential equations...just for clarity
 

Related Threads on Cauchy-Riemann condition

Replies
12
Views
5K
Replies
5
Views
5K
Replies
2
Views
2K
Replies
9
Views
723
Replies
18
Views
2K
  • Last Post
Replies
4
Views
923
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
4
Views
1K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
1
Views
2K
Top