Question

2-11 CAUCHY-RIEMANN EQUATIONS Are the following functions analytic? Use (1) or (7). 2. f(z) = izz 3. f(z) = e -2,0 (cos 2y – i sin 2y) 4. f(x) = e« (cos y – i sin y) 5. f(z) = Re (z?) – i Im (32) 6. f(x) = 1/(z – 25) 7. f(x) = i/28 8. f(z) = Arg 2TZ 9. f(z) = 3772/(23 + 4722) 10. f(x) = ln [z] + i Arg z 11. f(z) = cos x cosh y – i sin x sinh y

9 and 11 please

Answer 1

9. Let
2= 1 +iy
. Then
- 37² 1 37² (23 + 472x) =(22+ 472) (2+iy) ((x + iy)2 + 472) 372 (2 – iy) 37°C.x – iy) (22 - y2 + 472 – 2iry) (x2 + y2) (22 - y2 + 472 + 2ixy) (22 + y2)((x2 - y2 + 472)2 + 4.x2y2) 37² (0:23 – Ty? + 47²1 – 2.cy?) +i(y3 – x²y – 47'y – 2.rạy)) (x2 + y2)(x+ + y4 + 1674 +2.c2y2 + 872,2 – 87+2y2) 372((:23 – 3.ry2 + 472.0) +ily3 – 3.rºy – 47²y)) 7:6 + rºy4 + 16742 + 2r4y2 +87274 – 87?r?y2 + x4y2 + y + 1674y2 + 2.c2y4 + 872y2 – 87+y4 31((:23 – 3.ry2 + 4721) +ily3 – 3.rºy – 47ºy)) 76 +3.cy4 + 167472 +3.r4y2 +872,4 + y + 1674y2 – 87244

If we write
f(2) = u(x, y) + ivez,y)
then
3л (3 — 3х2 + 4х2х) * + 3х2у + 16n+2 + 3x4 y? + 8724 +y+ 164у2 — 8Ру
and
31 (7} – 3.ry – 47y) 26 +3.cº+ 1674,2 + 3.x+y2 + 872.,4 + y + 1674y2 - 87274
.

Now it we can check the Cauchy Riemann equations. Also in another way, has poles at z=0, z=2π and z=-2π. Hence f is not analytic at these points.

کی (2) = (2 + الاج) علي

11. Here,
ull,y) = cos x cosh y
and
var, y) = -sin c sinh y
. Now,
du = -sin cosh y, a = cos x sinh y, a = - cos z sinh y,
and
ou = -sin x cosh y ay
. Here clearly the Cauchy Riemann equations are satisfied and hence f is analytic.