6. (a) Recall that sin z = ¹2; cos z = (i) Find real and imaginary parts of sin z and cos z, for z = x+iy, x, y ER. (ii) Prove that | sinz|2 + | cos z|? = 1+2sinh?y for z= 2+i,,y€ R. (b) Prove that ur z EC, (c) Let 1 2 CoS z u(x, y) = √3 2 )= sin 2 = cos $(2+3). y+k x² + (y-1)2; v(x, y) = x³+y³ x² + y² For which values of k are the Cauchy-Riemann equations satisfied? (d) Let ax, y) = x² + (y-1)². y³ - 7³ x² + y² for = x+iy #0 and u(0,0) = 0 = v(0,0). Let f(z) = f(x+iy) = u(x, y) + iv(x, y). Prove that the Cauchy-Riemann equations for f are satisfied at the origin. (14
6. (a) Recall that sin z = ¹2; cos z = (i) Find real and imaginary parts of sin z and cos z, for z = x+iy, x, y ER. (ii) Prove that | sinz|2 + | cos z|? = 1+2sinh?y for z= 2+i,,y€ R. (b) Prove that ur z EC, (c) Let 1 2 CoS z u(x, y) = √3 2 )= sin 2 = cos $(2+3). y+k x² + (y-1)2; v(x, y) = x³+y³ x² + y² For which values of k are the Cauchy-Riemann equations satisfied? (d) Let ax, y) = x² + (y-1)². y³ - 7³ x² + y² for = x+iy #0 and u(0,0) = 0 = v(0,0). Let f(z) = f(x+iy) = u(x, y) + iv(x, y). Prove that the Cauchy-Riemann equations for f are satisfied at the origin. (14
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:6. (a) Recall that sin z =
elz-e-iz
21
cos z =
(i) Find real and imaginary parts of sin z and cos z, for z = x+iy, x, y = R.
(ii) Prove that
|sinz|? + | cos z|2 =1+2sinh’y for z=2+iy,2,y€R.
(b) Prove that ur z EC,
(c) Let
cos 2-3 sin z = cos (2+).
√3
2
2
u(x, y) =
elz+e-iz
u(x, y) =
y+k
; v(x, y) =
x² + (y-1)²¹
For which values of k are the Cauchy-Riemann equations satisfied?
(d) Let
2³+y³
x2 + y2
v(x, y) =
x
x² + (y-1)2
y³ – 2³
x² + y²
for 2 = x+iy 0
and u(0,0) = 0 = v(0,0).
Let f(2)=f(x + y) = u(x.y) +iv(x, y). Prove that the Cauchy-Riemann equations for
f are satisfied at the origin. (4
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