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
-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
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
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
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
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
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
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
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
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
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
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
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
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