Integration in the Complex Plane

PowerPoint presentation on Integration in Complex Plane. A topic of Complex Analysis.

Transcript

1.  ntegration in the Complex Plane Property of Amit Amola. Should be used for reference and with consent. 1 Prepared by: Amit Amola SBSC(DU)
2. Defining Line Integrals in the Complex Plane 0 a z  N b z  1 z 2 z 1  3 z 2  3  N  1 N z  n z n  x y #.  −1 #. ℎ = =1 ( )( − −1 ) #. ℎ → ∞, ℎ ℎ ∆ = − −1 → 0 = = lim →∞ = lim →∞ =1 ( )( − −1 ) ( ℎ ℎ ℎ ) C Property of Amit Amola. Should be used for reference and with consent. 2
3. Equivalence Between Complex and Real Line Integrals Note that- So the complex line integral is equivalent to two real line integrals on C. Property of Amit Amola. Should be used for reference and with consent. 3
4. Review of Line Integral Evaluation t t -a a t -a a 1 t 2 t 3 t 1 N t  C n t x y 0 t f N t t  1 t 2 t 3 t 1 N t  n t … … 0 t f N t t  t A line integral written as is really a shorthand for: where t is some parameterization of C : or Example- Parameterization of the circle x2 + y2 = a2 1) x= a cos(t), y= a sin(t), 0 t(=) 2 2) x= t, y= 2 − 2, t0 = , tf = − a And x= t, y=− 2 − 2, t0 = −, tf = a } ( , ) Property of Amit Amola. Should be used for reference and with consent. 4
5. Review of Line Integral Evaluation, cont’d 1 t 2 t 3 t 1 N t  C n t x y 0 t f N t t  While it may be possible to parameterize C(piecewise) using x and/or y as the independent parameter, it must be remembered that the other variable is always a function of that parameter i.e.- (x is the independent parameter) (y is the independent parameter) ( , ) Property of Amit Amola. Should be used for reference and with consent. 5
6. Line Integral Example Consider- x y C  r z z It’s a useful result and a special case of the “residue theorem” . , ℎ : = , = , 0 ≤ ≤ 2 ⟹ = + = , ⟹ = Property of Amit Amola. Should be used for reference and with consent. 6
7. Cauchy’s Theorem x y C #. Cauchy′s Theorem: If z is analytic in R then = 0 #. First, note that if = = + , then = ( − ) + + ; ….. − ℎ: R- a simply connected region Construst vectors = − , … = + , dr = + , in the plane and write the above equation as: () = . + . = × . + × . , . × = . − 0 = − − = 0, . × = . 0 = − = 0 ⟹ = 0 Stoke’s theorem C.R. Condition C.R. Condition Property of Amit Amola. Should be used for reference and with consent. 7
8. Cauchy’s Theorem(cont’d) x y C Consider R- a simply connected region #. Cauchy′s Theorem: If z is analytic in R then = 0 #. The proof using Stoke′s theorem requires that u x, y , v x, y have continuous first derivatives and that C be smooth … #. But the Goursat proof removes these restrictions; hence the theorem is often called the Cauchy − Goursat theorem Property of Amit Amola. Should be used for reference and with consent. 8
9. Extension of Cauchy’s Theorem to Multiply-Connected Regions x y 1 C 2 C 1 c 2 c x y 1 C 2 C R- a multiply connected region R’- a simply connected region #. If is analytic in R then: 1,2 =? #. Introduce an infinitesimal- width “bridge” to make R into a simply connected region R’ 1−2−1+2 = 1 − 2 = 0 #. It shows that integrals are independent of path. (since integrals along 1 and 2 are in opposite directions and thus cancel each other) Property of Amit Amola. Should be used for reference and with consent. 9
10. Fundamental Theorem of the Calculus of Complex Variables a z b z 1 z 2 z 3 z 1 N z  C n z x y 1 z  2 z  3 z  N z  is a path independent in a simply connected region R = lim →∞ =1  ∆ , ∆ = − −1 ,  on C between zn−1 and zn Suppose we can find z such that ′ z = = ; hence  ≈ Δ Δ = − (−1 ) ∆ = − (path independent if is analytic on the paths from ) ⇒ Property of Amit Amola. Should be used for reference and with consent. 10
11. Fundamental Theorem of the Calculus of Complex Variables (cont’d) #. Fundamental Theorem of Calculus: Suppose we can find such that ′ = = : ⟹ = = − #. This permits us to define indefinite integrals: = +1 + 1 , sin = − cos , = , #. In general F(z)= 0   for arbitrary 0 #. All the indefinite integrals differ by only a constant. Property of Amit Amola. Should be used for reference and with consent. 11
12. Cauchy Integral Formula x y C R 0 z 0 C 1 c 2 c x y C R- a simply connected region #. is assumed analytic in R but we multiply by a factor 1 − 0 that is analytic except at 0 and consider the integral around () ( − 0 ) #. To evaluate, consider the path + 1 + 2 + 0 shown that encloses a simply − connected region for which the integrand is analytic on and inside the path: +1+2+0 () ( − 0 ) = 0 ⟹ () ( − 0 ) = − 0 () ( − 0 ) Property of Amit Amola. Should be used for reference and with consent. 12
13. Cauchy Integral Formula, cont’d     0 0 0 C C f z f z dz dz z z z z       0 z 0 C C Evaluate the 0 integral on a circular path, for r ⟶ 0 ⇒ ⇒ Cauchy Integral Formula The value of at 0 is completely determined by its values on ! Property of Amit Amola. Should be used for reference and with consent. 13
14. Cauchy Integral Formula, cont’d 0 z 0 C C 0 z C Note that if 0 is outside C, the integrand is analytic inside C, hence by the Cauchy Integral Theorem, 1 2 () − 0 = 0 In summary, () − 0 = 2 0 , 0 inside 0, 0 outside Property of Amit Amola. Should be used for reference and with consent. 14
15. Derivative Formula 0 z C Since () is analytic in C , its derivative exists; let’s express it in terms of the Cauchy Formula: 0 = 1 2 () − 0 . So ⇒ ⇒ ⇒ We’ve also just proved we can differentiate w.r.t. 0 under the integral sign! Property of Amit Amola. Should be used for reference and with consent. 15
16. Derivative Formulas, cont’d 0 z C If () is analytic in C, then its derivatives of all orders exist, and hence they are analytic as well. In general, Similarly or ⇒ Property of Amit Amola. Should be used for reference and with consent. 16