Question: Determine which of the following functions are analytic: 1. x + y 2. 2xy +i (x- y) 1 (z-1) (z+1) x-iy 5. x-iy +

Determine which of the following functions are analytic: 1. x + y 2. 2xy +i (x - y) 1 (z-1) (z+1) x-iy 5.

Determine which of the following functions are analytic: 1. x + y 2. 2xy +i (x- y) 1 (z-1) (z+1) x-iy 5. x-iy + a 6. sin x cosh y + i cos x sinh y 4. 460 Ans. Not analytic 7. xy + y sorrect answer: 3. Ans. Analytic at all points y = x Ans. Not analytic Ans. Analytic at all points, except z = 1 Ans. Not analytic Ans. Yes, analytic Scanned with CamScanner Funcion Ans. Yes, analytic at origin o ha continuous at z= 0, we should define f (0) equal

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1 x 2 iy 2 Cauchy Riemann Equations are UxVy and UyVx Ux 2 differentiate partially with respect to x Ux2x differentiate partially with respect to y Uy0 Vy 2 differentiate partially with respect to x Vx0 differentiate partially with respect to y Vy2y Cauchy Riemann equations are satisfied when yx This function is analytic 2 2xyix 2 y 2 U2xy differentiate partially with respect to x Ux2y differentiate partially with respect to y Uy2x V x 2 y 2 differentiate partially with respect to x Vx2x differentiate partially with respect to y Vy2y Cauchy Riemann equations are not satisfied Ux not equal to Vy and Uy not equal to Vx so this function is not analytic 3 xiyx2y 2 U xx2y21 Differentiating partially with respect x Uxx2xx2y22x2y21 Differentiating partially with respect to y Uyx1x2y22 2y V yx2y2 Vyx2y21 Differentiating partially with respect x Vxyx2y22 2x Differentiating partially with respect to y Vy x2y2 1 y2y x2y22 Vy x2y2 x2y22 Cauchy Riemann equations are not satisfied Ux not equal to Vy and Uy not equal to Vx This function is not analytic 4 1z1 z1 1z21 The denominator must be equal to zero z210 z or 1 So This is analytic in all points except z or 1 5 xiyxiya xiyxaiy xaiyxaiy xiyxa iyxiyxa2y2 U xiyxaxa2y2 x2axiyxaiyxa2y2 differentiate partially with respect to x Ux xa2y2 2xaiy x2axiyxaiy 2xa xa2y22 differentiate partially with respect to y Uy ... View full answer

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