Tham khảo tài liệu 'advanced mathematical methods for scientists and engineers episode 2 part 4', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Cauchy s Integral Formula Result Cauchy s Integral Formula. If f Z is analytic in a compact closed connected domain D and z is a point in the interior of D then f z -C 4 f ZL dz -C v f dz. 12n JdD z z 12n k J k z - z Here the set of contours Ck make up the positively oriented boundary dD of the domain D. More generally we have fw z _nL f z dZ _n_ V -fl dZ. f z - L Z z dZ 2 ỲẤ Z z n 1 dZ. Cauchy s Formula shows that the value of f z and all its derivatives in a domain are determined by the value of f z on the boundary of the domain. Consider the first formula of the result Equation . We deform the contour to a circle of radius Ỗ about the point z z. fizldi f z - f z dZ Jcs z - z z z We use the result of Example to evaluate the first integral. f dZ I2nf Z CỖ f Zf dZ z z 494 The remaining integral along Cs vanishes as Ỏ 0 because f z is continuous. We demonstrate this with the maximum modulus integral bound. The length of the path of integration is 2nd. lim i - dz limf 2nd - max If Z f z T-0 Jcs z z T-oV ỗ z-z J lim 2n max If z f z I T-0 Í-Z Ổ 0 This gives us the desired result. f z i2n JC ff dz z z dZ We derive the second formula Equation from the first by differentiating with respect to z. Note that the integral converges uniformly for z in any closed subset of the interior of C. Thus we can differentiate with respect to z and interchange the order of differentiation and integration. f n z -1 d f z f i2n dzn JC z z -L ị ẫ f ZL dz i2n C dzn z z _n_ i f z dz 12n Jc z z n 1 Example Consider the following integrals where C is the positive contour on the unit circle. For the third integral the point z 1 is removed from the contour. 2. ỈC z 3 13z 1 dz .
