Downloaded 158 times
More Related Content
Similar to Cauchy integral theorem & formula (complex variable & numerical method )
More from Digvijaysinh Gohil
Cauchy integral theorem & formula (complex variable & numerical method )
- 1. Subject :- ComplexVariable & Numerical Method (2141905) Topic :- Cauchy Integral Theorem & Formula Branch :- Mechanical Semester :- 4th •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, c 0dz)z(f
- 3. Examples 1. Evaluate wherec 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. c 4z dz 4z 1 )z(f
- 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, 0 4z dz c
- 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 | c z i2z e 15 14
- 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, c z 0 i2z e
- 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, c zdzsec zcos 1 zsec)z(f ,.... 2 5 , 2 3 , 2 z 0zdzsec c
- 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, • 𝐹 𝑎 = 1 2𝜋𝑖 𝐹(𝑥) 𝑧−𝑎 . 𝑑𝑧
- 9. Cauchy Integral FormulaFor 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 • 𝐹′ 𝑎 = 1 2𝜋𝑖 𝐹(𝑥) (𝑧−𝑎)2 . 𝑑𝑧 • In general, • 𝐹 𝑛 𝑎 = 𝑛! 2𝜋𝑖 𝐹(𝑥) (𝑧−𝑎) 𝑛+1 . 𝑑𝑧
- 10. Ex – 1:- 𝒛 𝒛−𝟐 . 𝒅𝒛 , where C is 𝒛 − 𝟐 = 𝟑 𝟐 • Solution :- • Here I = 𝑧 𝑧−2 . 𝑑𝑧 • And C is a circle with center (2,0) and radius 3 2 • 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, • 𝐹 𝑎 = 1 2𝜋𝑖 𝐹(𝑍) (𝑍−𝑎) . 𝑑𝑧
- 11. Cont... • 𝐹(𝑧) (𝑧−𝑎) . 𝑑𝑧 =2𝜋𝑖 𝐹(𝑎) • 𝑧 𝑧−2 . 𝑑𝑧 = 2𝜋𝑖 𝐹(2) • 𝑧 𝑧−2 . 𝑑𝑧 = 2𝜋𝑖 2 • 𝑧 𝑧−2 . 𝑑𝑧 = 4𝜋𝑖
- 12. Ex – 2:- 𝒅𝒛 (𝒛 𝟐−𝟕𝒛+𝟏𝟐) , C is |Z| = 3.5 • Let 𝐼 = 𝑑𝑧 (𝑧2−7𝑧+12) • 𝐼 = 𝑑𝑧 (𝑧−4)(𝑧−3) • 𝐼 = 1 (𝑧−4) (𝑧−3) . 𝑑𝑧 • Here, C is a circle with center (0,0) and radius 3.5 • Z=3 lies inside C, so let F(z) = 1 (𝑧 − 4) • F(z) is analytic everywhere inside and on C, • Now, by Cauchy Integral Formula,
- 13. Cont... • 𝐹(𝑧) (𝑧−𝑎) . 𝑑𝑧 =2𝜋𝑖 𝐹(𝑎) • 𝑑𝑧 (𝑧2−7𝑧+12) = 2𝜋𝑖 𝐹(3) • 𝑑𝑧 (𝑧2−7𝑧+12) = 2𝜋𝑖 1 3−4 • 𝑑𝑧 (𝑧2−7𝑧+12) = −2𝜋𝑖
- 14. Ex – 3:- 𝒔𝒊𝒏𝝅𝒛 𝟐+𝒄𝒐𝒔𝝅𝒛 𝟐 𝒛−𝟏 (𝒛−𝟐) . 𝒅𝒛 where C is |z| = 3 • Here, C is a circle with (0,0) and radius 3 • Now, 1 𝑧−1 (𝑧−2) = 𝐴 𝑧−1 + 𝐵 (𝑧−2) • By partial fraction, • 1 = 𝐴 𝑧 − 2 + 𝐵 𝑧 − 1 • For 𝑧 = 2, 𝐵 = 1 • For 𝑧 = 1, 𝐴 = −1 • 𝑠𝑖𝑛𝜋𝑧2+𝑐𝑜𝑠𝜋𝑧2 𝑧−1 (𝑧−2) . 𝑑𝑧 = − 𝑠𝑖𝑛𝜋𝑧2+𝑐𝑜𝑠𝜋𝑧2 𝑧−1 . 𝑑𝑧 + 𝑠𝑖𝑛𝜋𝑧2+𝑐𝑜𝑠𝜋𝑧2
- 15. Cont... • Here, Cis a circle with center (0,0) and radius 3 • Z=1 and 2 lies inside C, so let F(z) = 𝑠𝑖𝑛𝜋𝑧2 + 𝑐𝑜𝑠𝜋𝑧2 • F(z) is analytic everywhere inside and on C, • Now, by Cauchy Integral Formula, • 𝐹 𝑎 = 1 2𝜋𝑖 𝐹(𝑧) (𝑧−𝑎) . 𝑑𝑧
- 16. Cont... • 𝐼 =−2𝜋𝑖 𝐹 1 + 2𝜋𝑖 𝐹(2) • 𝐼 = −2𝜋𝑖 0 − 1 + 2𝜋𝑖 0 + 1 • 𝐼 = 2𝜋𝑖 + 2𝜋𝑖 • 𝐼 = 4𝜋𝑖
- 17. Ex – 4:- 𝐳 (𝐳−𝟏) 𝟑 . 𝐝𝐳, C is |z|=2 • Let 𝐼 = z (z−1)3 . dz • 𝐼 = z (z−1)2+1 . dz • 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... • z (z−1)3 . dz= 𝐹′′ 1 × 2𝜋𝑖 2! • 𝐹 𝑧 = 𝑧 • 𝐹′ 𝑧 = 1 • 𝐹′′ 𝑧 = 0 • z (z−1)3 . dz = 0 × 𝜋𝑖 • z (z−1)3 . dz = 0