Question

17. Suppose that $f(z)$ and $g(z)$ are analytic on a region D. Moreover, C is a curve on D and the starting and terminal points of C are $z_1$ and $z_2$, respectively. Prove that $$\int_C f(z)g'(z)dz = f(z)g(z)\Big|_{z_1}^{z_2} - \int_C f'(z)g(z)dz.$$18. Use the formula in Problem 17 to determine $$\int_C z \sin(z)dz,$$ C: $z(t) = \sin(\pi t^2) + i\cos(\pi t/2)$ where $-1 \le t \le 1.$

          17. Suppose that $f(z)$ and $g(z)$ are analytic on a region D.
Moreover, C is a curve on D and the starting and terminal points of C are $z_1$ and $z_2$, respectively. Prove that
$$\int_C f(z)g'(z)dz = f(z)g(z)\Big|_{z_1}^{z_2} - \int_C f'(z)g(z)dz.$$18. Use the formula in Problem 17 to determine
$$\int_C z \sin(z)dz,$$
C: $z(t) = \sin(\pi t^2) + i\cos(\pi t/2)$ where $-1 \le t \le 1.$

17. Suppose that f(z) and g(z) are analytic on a region D.
Moreover, C is a curve on D and the starting and terminal points of C are z1 and z2, respectively. Prove that

f(z)g'(z)dz = f(z)g(z)|z1^z2 - f'(z)g(z)dz.
18. Use the formula in Problem 17 to determine

z sin(z)dz,

C: z(t) = sin(π t^2) + icos(π t/2) where -1 ≤ t ≤ 1.

Added by Kristin M.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Raj Bala

01/31/2022

Step 1

Step 1: Apply the formula from Problem 17, which states that \int_C f(z)g^(')(z)dz=f(z)g(z)|_(z_(1))^(z_(2))-\int_C f^(')(z)g(z)dz. Show more…

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please answer problem18, Suppose that f(z) and g(z) are analytic on a region D. Moreover, C is a curve on D and the starting and terminal points of C are z_(1) and z_(2), respectively. Prove that int_C f(z)g^(')(z)dz=f(z)g(z)|_(z_(1))^(z_(2))-int_C f^(')(z)g(z)dz.|| Use the formula in Problem 17 to determine int_C zsin(z)dz C:z(t)=sin(pi t^(2))+icos(pi (t)/(2)) where -1<=t<=1. 17. Suppose that f(z) and g(z) are analytic on a region D. Moreover, C is a curve on D and the starting and termi- nal points of C are zi and z2, respectively. Prove that 71 zp(z f(z)g'(z)dz=f(z)g(z) 18. Use the formula in Problem 17 to determine z sin(z)dz, JC C:z(t=sin(Tt+ icos(Tt/2 where-1t1.