How to Show u(x,y) and v(x,y) are Constant Throughout D?

In summary, if v is a harmonic conjugate of u in a domain D and u is a harmonic conjugate of v in D, then u(x,y) and v(x,y) are constant throughout D. This can be shown by using the properties of harmonic conjugates, where u_x=v_y and u_y=(-v_x) in one direction, and v_x=u_y and v_y=(-u_x) in the other direction. Putting these together, it can be seen that both u and v satisfy the Laplace's equation u_xx + u_yy = 0 and v_xx + v_yy = 0, which means they are both constant throughout D. Therefore, u(x,y) and v(x,y) are constant
  • #1
tylerc1991
166
0

Homework Statement



Suppose v is a harmonic conjugate of u in a domain D, and that u is a harmonic conjugate of v in D. Show how it follows that u(x,y) and v(x,y) are constant throughout D.

The Attempt at a Solution



since u is a harmonic conjugate of v, u_xx + u_yy = 0
also, since v is a harmonic conjugate of u, v_xx + v_yy = 0

u_x = v_y => u_xx = v_yx
u_y = -v_x => u_yy = -v_xy

I think that I am going in circles here. Can someone lend a helping hand with this problem? Thank you so much!
 
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  • #2
Hint: in one direction u_x=v_y and u_y=(-v_x). In the other direction v_x=u_y and v_y=(-u_x). Put them together.
 
  • #3
what do you mean in one direction?

do you mean that since u is a harmonic conjugate then that is one direction, and since v is a harmonic conjugate then that is the other direction?
 
  • #4
tylerc1991 said:
what do you mean in one direction?

do you mean that since u is a harmonic conjugate then that is one direction, and since v is a harmonic conjugate then that is the other direction?

Yes, that's exactly what I mean.
 

Related to How to Show u(x,y) and v(x,y) are Constant Throughout D?

1. What is Complex Analysis?

Complex Analysis is a branch of mathematics that studies functions of complex numbers. It involves understanding the behavior of these functions, their properties, and their applications in various areas of mathematics and physics.

2. What are complex numbers?

Complex numbers are numbers that involve both a real and an imaginary component. They are written in the form a + bi, where a and b are real numbers and i is the imaginary unit (√-1).

3. What are the main applications of Complex Analysis?

Complex Analysis has a wide range of applications, including solving differential equations, understanding the behavior of electric and magnetic fields, and analyzing fluid flow. It is also used in signal processing, quantum mechanics, and many other areas of mathematics and physics.

4. What are the key concepts in Complex Analysis?

Some key concepts in Complex Analysis include analytic functions, Cauchy-Riemann equations, contour integration, and the Cauchy integral theorem. These concepts are used to understand the behavior and properties of complex functions.

5. How is Complex Analysis different from Real Analysis?

Complex Analysis deals with functions of complex numbers, while Real Analysis deals with functions of real numbers. Complex Analysis also involves studying the properties of complex functions, such as differentiability and analyticity, which differ from the properties of real functions. Additionally, Complex Analysis has its own set of theorems and techniques that are specific to the study of complex functions.

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