Cauchy integral formula.pptx

Cauchy integral formula and Poisson integral formula

1. CHAUDHARY BANSI Lal university Cauchy’s Integral Formula And Poisson Integral Formula PRESENTED TO: PRESENTED BY: DEPARTMENT OF RASHMI MATHEMATICS M.Sc (MATHEMATICS) 2 nd SEM 220000603038

2. CAUCHY’S INTEGRAL FORMULA Statement:- If f(z) is analytic function within and on a closed contour C, and if is any point inside C then f( ) = dz Proof :- To prove the formula describe a circle V about the center of small radius r such that this circle |z- |=r does not intersect the curve C. The function is analytic in the region bounded by C and V. Then from Cauchy theorem , we have dz = dz …………….(1) dz = dz + dz ……………..(2) Since f(z) is analytic within C and so it is continuous at z = So , that given >0 , there exits 0 such that |f(z) - f( )|= …( 3) for |z- |= ……………..(4)  

3. Since r is at our choice and we can take r < so that (4) is satisfied for all z on the circle V For any point z on V , z- = r hence dz = r d dz = r d = 2 f( ) Hence by (2) dz - 2 f( ) = dz dz = 2 = 2 Since , is arbitrary small making dz - 2 f( ) = 0 f( ) = dz Hence the proof of the theorem  

4. Example:- Using Cauchy integral formula calculate dz , where C is the circle z =2 described in the positive sense   Solution :- Let f(z) = Therefore z = 3 and 3 lies outside the circle hence the function is analytic within and on the contour C And z = - lies inside C Hence by Cauchy integral formula , f( ) = dz f( - ) = dz = 2 = 2 [ ] = 2 [ ] =  

5. Some More Extensions Of Cauchy Integral Formula Cauchy integral formula for multiply connected region:- Statement :- if f(z) is analytic in a ring shaped bounded by two closed contours C₁ and C₂ and is a point in the region between C₁ and C₂ . Then f( ) = dz - dz where C₂ is the outer curve Cauchy integral formula for the derivative of an analytic function:- Statement :- if f(z) is analytic within and on a closed contour C and is any point lying in it . Then ( ) = dz Cauchy integral formula for the higher order derivative of an analytic function:- Statement :- if f(z) is analytic within and on a closed contour C and is any point within C . Then derivative of all orders are analytic and are given by ( ) = dz  

6. Example :- evaluate dz where c is z =3   Solution :- we know that ( ) = dz ……………(1) Let f(z) = is analytic within and on C and z = -1 lies inside C . Taking n = 3 in (1) (-1) = dz …… (2) f(z) = , (z) = 2 , (z) = 4 , (z) = 8 therefore (-1) = 8 , put this value in (2) 8 = dz dz =  

7. POISSON INTEGRAL FORMULA STATEMENT :- let(z) be analytic in the region |z| ≤ R , Then for 0 < r < R we have f(r ) = d where is the value of on the circle |z|= R PROOF :- Let C denote the circle |z|= R and let = r , 0 < r < R be any point inside C . Then by Cauchy’s integral formula f( ) = dz …………………(1) The inverse of w.r.t the circle |z|= R is and lies outside the circle So by Cauchy’s theorem , we have 0 = dz …………………………(2) Subtracting (2) from (1) , we get f( ) = dz ……………………..(3)  

8. Now any point on the circle C is expressible as z =R and also = r so = r Therefore - = - ……………..(4) Now )( - ) = z - - +z = - r -r +R = R [ - Rr -r R + ] = R [ - Rr( + ) + ] = R [ - Rr( + ) + ] = R [ - Rr(cos )- sin )+cos )+ sin ))+ ] = R [ - 2Rrcos )+ ] ………..(5) Thus equation (3) becomes f(r ) = d f(r ) = d Which is the required formula  

9. REFRENCES H.A Priestly , introduction to complex analysis Real and Complex analysis by Walter Rudh Complex analysis by H.D Pathak

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