(5) Use Cauchy Integral Formula to calculateh(2+(i+1), ee is whose vertices are 0, 4, 2- 2i,...
6. (1 point) Use Stokes' Theorem to find the line integral /2y dx + dy + (4-3x) dz, where C is the boundary of the triangle with vertices (0,0,0), (1,3,-2), and -2,4,5), oriented counterclockwise as viewed from the point (1, 0, 0) 6. (1 point) Use Stokes' Theorem to find the line integral /2y dx + dy + (4-3x) dz, where C is the boundary of the triangle with vertices (0,0,0), (1,3,-2), and -2,4,5), oriented counterclockwise as viewed from the...
Evaluate the integral $ 3z + 4)cos z dz, C:12+2i = 1 counterclockwise. z² +4 Integrate $c +436 yosz dz, C:/z+3–2i = 3 counterclockwise. (z +4)
1 5. Let A = dz, (2 – 1)2(2 + 2i)3 where I is the circle [2] = 3 traversed once counterclockwise. The following is an outline of the proof that A = 0, justify each statement. Jo Tz – 1)*(x + 2133 (a) For R > 3 show that A = A(R) where A(R) Som 1 (z – 1)2(x + 2i)3 dz, and I'R is the circle (2|| = R traversed once counterclockwise. 21R (b) For R > 3...
2-1/2dz if C is a polygonal line with vertices 2,1 + i,-1 i,-2 (without the segment [-2,2) and z-1/ is a principal value. Hint: consider a particular branch which is analytic on the contour uate the following integrals (all contours are positively oriented): cosh(z) 3 dz if C is a square of vertices 1 ti,-1ti C 2 sin(2) dz if C is a circle 3 2(2,2 2 3 dz if C is a rectangle with sides along the lines x-1,x--1,y...
4. (a) Use Cauchy's residue Theorem to provide alternative proofs for Cauchy's integral formula and its extension. (b) Evaluate z+ 4z + 5 dz, son z2 + z where C is the positively oriented circle of radius 2 centered at the origin.
-. Use Cauchy Integral Formula (and/or its higher-order extension) to calculate the following integrals over given circles. You may have to change the contour of integration into a set of closed curves surrounding each singularity inside the contour, as we did in class: (a) fja-2 522 (b) fiel-5 in 4, dz (c) fz–21=2 (22_1)2 dz (d) $jz+2+il+2 3+2ja dz 532+2z+1 dz
Question 1) Find I = z +2 3z - 2 + 3i 22 + (2i - 2)2 - 4i ] dz, C:\z| = 3, CW a. 4πί b. 8πί C. 2πί d. -2π(3 +i) e. 0.0 f. ο g. -4πί h. 6π
Evaluate the integral 5. Ten dz, where C is the boundary of the square with vertices at the points 0, 1, 1+i and i, with a counter clockwise orientation. What is the integral over the reverse contour?
5(a)(b) are asking what the Cauchy-Goursat Theorem and the general Cauchy Integral Theorem talks about. Please use these two theorems to solve the problem. (6) Let C denote the closed contour (3 – sint)et, 0 <t < 2n. Use 5(a)(b) above to aid in computing the following contour integrals. (a) So z?sin(2)dz (b) Jc E-P-5)² dz 24-iz
2. More integrals! Evaluate each integral, using either a Cauchy Integral Formula or Cauchy's Residue Theorem. Take C to be the circle [2] = 3, oriented counter-clockwise. 1) Sota-1jad: 6) Se TH h) Sorºcos(1/2)da