Cauchy integral theorem & formula (complex variable & numerical method )

complex variable & numerical method

1. Subject :- Complex Variable & Numerical Method ( 2141905) Topic :- Cauchy Integral Theorem & Formula Branch :- Mechanical Semester :- 4 th Made By :- - Divyangsinh Raj (150990119004) - Naved Fruitwala (150990119006) Utkarsh Gandhi (150990119007) Guided By :- Dr . Purvi Naik

2. Cauchy Integral Theorem If f(z) is analytic function and f '(z) is continuous at each point inside and on closed curve c then,

3. Examples 1. Evaluate where c is circle |z| = 2. f (z) is not analytic at z = - 4. Here c is |z| = 2 which is a circle with center (0,0) and radios 2.

4. Here z = - 4 lies outside the curve c. Therefore f'(z) is analytic inside and on c and also f ' (z) is continuous in z plane. Therefore by Cauchy integral theorem,

5. 2. Evaluate ; |c | : |z-1|=1 c is circle with center (1,0) and radios 1. f(z) is not analytic at z=-2i For |z-1|= |-2i-1 |

6. z = - 2i lies outside to curve c. So f(z) is analytic on and inside c. f ' (z) is continuous on and inside c. Therefore by Cauchy integral theorem,

7. 3. Evaluate ; |c| : |z|=1 f (z) is not analytic at cos z = 0. i.e. Now c is circle with center (0,0) and radios 1. These points lie outside c. So the f(z) is analytic and f(z) is continuous. So by Cauchy integral theorem,

8. Cauchy Integral Formula If a function F(a) is analytic inside a closed curve C and if a is any point inside C then,  

9. Cauchy Integral Formula For Derivative If a function F(x) is analytic in region R then its derivative at any point z=a is also analytic in R and it is given by In general,  

10. Ex – 1 :- , where C is   Solution :- Here I = And C is a circle with center (2,0) and radius Here z=2 lies inside C, so let F(z) = z F(z) is analytic everywhere inside and on C, Now, by Cauchy Integral Formula,  

11. Cont...  

12. Ex – 2 :- , C is |Z| = 3.5   Let Here, C is a circle with center (0,0 ) and radius 3.5 Z=3 lies inside C, so let F(z) = F(z) is analytic everywhere inside and on C, Now, by Cauchy Integral Formula,  

13. Cont...  

14. Ex – 3 :- where C is |z| = 3   Here, C is a circle with (0,0) and radius 3 Now, By partial fraction, For For  

15. Cont... Here, C is a circle with center (0,0 ) and radius 3 Z=1 and 2 lies inside C, so let F(z) = F(z) is analytic everywhere inside and on C, Now, by Cauchy Integral Formula,  

16. Cont...  

17. Ex – 4 :- , C is |z|=2   Let Here F(z)=z and a=1 which is on C Here, C is a circle with center (0,0) and radius 2 F(z) is analytic everywhere inside and on C, Now by Cauchy Integral Formula For Derivative,  

18. Cont...