Lecture 17 Complex Analysis

Polar form of C-R equations:-
We can write C-R eq. by polar form such that
u_r=1/r v_? & u_?=-rv_r
2.8 Harmonic Functions:-
Let h(x,y) be a real function depend on two variable x,y and it has domain in xy-plane.we are said to be a function h is harmonic function in this domain, if the first and second partial derivatives of its are continuous function and satisfied this equation.
(?^2 h)/??x?^2 +(?^2 h)/(?y^2 )=0
We are said to be this equation by Laplace s equation we can rewrite equation (1) as follow:-
?^2 h=0 Where ?^2=?^2/(?x^2 )+?^2/(?y^2 )
EX:-Prove that f(x,y)=e^x cos?y is harmonic fun.
?f/?x=e^x cos?y , ?f/?y=-e^x sin?y
(?^2 f)/(?x^2 )=e^x cos?y ,(?^2 f)/(?y^2 )=-e^x cos?y
(?^2 f)/(?x^2 )+(?^2 f)/(?y^2 )=e^x cos?y+(-e^x cos?y)
=e^x cos?y-e^x cos?y=0
Theorem:- If f(Z)=u(x,y)+iv(x,y) is analytic functions in region D then u,v are harmonic functions in region .

2.9 Harmonic Conjugate:-
I f the two function u(x,y) and v(x,y) are harmonic fun. In region D then f(Z)=u(x,y)+iv(x,y) is an analytic function in this region and v(x,y) is said to be a harmonic conjugate of the function u .










Remark:-
If f(Z) is a harmonic (analytic) function then
u_xx+u_yy=0 & v_xx+v_(yy=0)
The word "conjugate" in above definition is different conjugate of the complex number Z ?.
If v(x,y) a harmonic conjugate of u(x,y) , then -u(x,y) is a harmonic conjugate of v(x,y) where the functions v(x,y) and


-u(x,y) are real and imaginary parts of function-if(z),that is v(x,y) is a h.c.(harmonic conjugate) of u(x), then
f(Z)=u(x,y)+iv(x,y) and -u(x,y) is a h.c. Of v(x,y) , then -if(Z)=-i(u(x,y)+iv(x,y) )=v(x,y)-iu(x,y).