Question

find the analytic function w=u+iv given v=e-2xy sin(x2-y2)

Answer 1

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Solution:-

Let f = u + iv. Then,

u = x

2 − y

2 − 2y ⇒ ux = 2x = vy ⇒ v = 2xy + φ(x) ⇒

−vx = −2y − φ

0

(x) = uy = −2y − 2 ⇒ φ(x) = 2x + c ⇒

f = x

2 − y

2 − 2y + i(2xy + 2x + c)

where c is a real constant.

Let g = u + iv. We have:

v = 2xy + y ⇒ vy = 2x + 1 = ux ⇒ u = x

2 + x + φ(y) ⇒

uy = φ

0

(y) = −vx = −2y ⇒ φ(y) = −y

2 + C ⇒

g = x

2 − y

2 + x + c + i(2xy + y)

where as before c is a real constant.

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Answer 2

Answer:

e^-iz^

Step-by-step explanation: