[tex]\huge{\bf{{\underline{\colorbox{Orange} {\color{white}{Question}}}}}}[/tex]
define hormonic conjugate
with formula
Share
[tex]\huge{\bf{{\underline{\colorbox{Orange} {\color{white}{Question}}}}}}[/tex]
define hormonic conjugate
with formula
Sign Up to our social questions and Answers Engine to ask questions, answer people’s questions, and connect with other people.
Login to our social questions & Answers Engine to ask questions answer people’s questions & connect with other people.
[tex]{\mathcal{\red{\huge{answer}}}}[/tex]
*Definition of Harmonic Conjugate*
In mathematics, two points A and B are said to be *harmonic conjugates* of each other with respect to another pair of points C and D if the cross ratio (ABCD) equals 1 [].
*Harmonic Conjugate in Complex Analysis*
In the context of complex analysis, the concept of harmonic conjugate is closely related to harmonic functions. If f(z) = u + iv is an analytic function, then v + iu is said to be the harmonic conjugate of u + iv, where u and v are real-valued functions. Both u and v are harmonic functions, meaning they satisfy the Laplace's equation [].
*Formula for Harmonic Conjugate*
To find a harmonic conjugate, we can use the Cauchy-Riemann equations. Let's consider a function f(z) = u(x, y) + iv(x, y), where z = x + iy. The Cauchy-Riemann equations are given by:
- ∂u/∂x = ∂v/∂y
- ∂u/∂y = -∂v/∂x
By solving these partial differential equations, we can find the harmonic conjugate v(x, y) of the given function u(x, y) [].
It's worth noting that the formula for the harmonic conjugate may vary depending on the specific context and problem at hand. The formula mentioned above is a general approach for finding the harmonic conjugate in complex analysis.
I hope this explanation clarifies the concept of harmonic conjugate for you. Let me know if you have any further questions!
Verified answer
[tex]\huge\mathcal{\fcolorbox{maroon} {maroon} {\red{ᏗᏁᏕᏇᏋᏒ}}}[/tex]
two points A and B are said to be *harmonic conjugates* of each other with respect to another pair of points C and D if the cross ratio (ABCD) equals 1 [].
▬▬▬▬▬▬▬▬▬▬▬▬▬▬
To find a harmonic conjugate, we can use the Cauchy-Riemann equations. Let's consider a function f(z) = u(x, y) + iv(x, y), where z = x + iy. The Cauchy-Riemann equations are given by:
- ∂u/∂x = ∂v/∂y
- ∂u/∂y = -∂v/∂x
By solving these partial differential equations, we can find the harmonic conjugate v(x, y) of the given function u(x, y) [].
▬▬▬▬▬▬▬▬▬▬▬▬▬▬