The document discusses Cauchy Riemann equations, including its history, important features, definition, and applications. It was discovered in 1851 by Augustin Cauchy and Bernhard Riemann during work on the theory of functions. The equation is used to check the differentiability and analyticity of complex functions. It has applications in engineering fields like triangular grid generation for computational fluid dynamics simulations. It also has applications in verifying Maxwell's equations and calculating fluid intensity and divergence.
5. IMPORTANT FEATURES
Cauchy Riemann equation also known as
D”Alembret Euler condition
It is use to check differentiability of a complex
function
It is used for verification of the analicity of the given
function
It is used to find the harmonic conjugate of a
function
6. CAUCHY RIEMANN EQUATION
• Definition
if f(z)=u(x,y)+i(x,y) and x=rcos(theta) ; y=rsin(theta)
Therefore u and v are the function of r and theta.
Then Cauchy Riemann equation Cartesian co-ordinate is given as
1. du/dx =dv/dx ; 2. dv/dx= -du/dx
in polar form
1. du/dr=1/r(dv/d*) 2. dv/dr= -1/r(du/d*)
8. DRAMATIC DISCOVERY OF
CAUCHY RIEMANN EQUATION
• Aungestin Cauchy was a civil engineer and mathmation also..
• During his survey for the dam he got this idea ……..
9. APPLICATION IN ENGINEERING
• Triangular grid generation
According to the research paper of Hiroaki nishkiwa on computational
fluid dynamics ……. Department of aerospace engineering university of
Michigan(30th march 1999)
12. OTHER APPLICATION
• Following are the other application driven with help of Cauchy Riemann
equation are……………….
• Calculation of fluid intensity at a point in the fluid
• For the verification of Maxwell equation
• In divergence theorem to give the rate of change of a function
13. APPLICATION IN REAL LIFE
• To move point a function from Cartesian to polar co-ordinate by
D.Alembret Euler condition
14. OTHER APPLICATION
• To prove function analytic
• To find harmonic conjugate of a function
