2(b) Given the function: u(x.y) = exp(-x) [x sin y – y cos
y]

Find v(x,y) such that f(z) = u + iv is analytic

Solution:

a) ¶ u/ ¶ x = exp(-x) (sin y) - x exp(-x) sin y + y (exp(-x) ) cos y

**=****¶ v/ ¶ y**
b) - ¶ u/ ¶ y = - x exp (-x) cos y
- y (exp(-x) sin y +

exp(-x) cos y

exp(-x) cos y

**=****¶ v/ ¶ x**
Integrate
(a) with respect to y,

*keeping x constant*so that:
v =

**-**exp(-x) cos y + x exp (-x) cos y**-**exp(-x)[ y sin y + cos y) + F(x)
v =

**y**exp(-x) sin y + x exp(-x) cos y + F(x)
Here, F(x) is an arbitrary real function of x.

Substituting
the last result for v into the Cauchy equation for

¶ v/ ¶ x we get:

¶ v/ ¶ x we get:

y exp(-x) sin y – x exp(-x) cos y + exp (-x) cos y + F’(x)

= - y exp (-x) sin y
– x exp (-x) cos y – y exp(-x) sin y

Or: F’(x) = 0 and
F(x) = c (constant) Then, from the earlier
expression for v:

v = exp (-x) (y sin
y + x cos y) + c

3) Let f(z) = exp(x) cos(y) + i(exp(x)sin(y) = u(x,y) + iv(x,y)

a) Determine if the function is analytic for both u and v.

b) Determine if the function is harmonic for both u and v

a) Determine if the function is analytic for both u and v.

b) Determine if the function is harmonic for both u and v

Solutions:

a)
¶ u/ ¶ x
= exp (x) cos y

And: ¶
v/ ¶ y = exp (x) cos y
so: ¶ u/ ¶ x
= ¶ v/ ¶ y

SO the function is analytic for u.

Now, ¶ v/ ¶ x = exp(x) sin y and :

- ¶ u/ ¶ y = - exp
(x) [- sin y] = exp(x) sin y

SO: ¶ v/ ¶ x = - ¶ u/ ¶ y

Therefore, the function is also analytic for v.

b) ¶

^{2}u/ ¶ x^{2}= ¶ / ¶ x [ exp (x) cos y ] = exp (x) cos y
¶

^{2}u/ ¶ y^{2}= ¶ / ¶ y [- exp (x) sin y ] = - exp(x) cos y
Then: ¶

(- exp(x) cos y) = 0

^{2}u/ ¶ x^{2}+ ¶^{2}u/ ¶ y^{2}= exp (x) cos y +(- exp(x) cos y) = 0

So
the function is harmonic for u.

__Looking now at v__:

¶

^{2}v/ ¶ x^{2}= ¶ / ¶ x [exp(x) sin y ] = exp(x) sin y
¶

=

^{2}v/ ¶ y^{2}= ¶ / ¶ y [exp (x) cos y ] = exp (x) [- sin (y)]=

**-**exp (x) sin y
Then: ¶

= exp(x) sin y + (

^{2}v/ ¶ x^{2}+ ¶^{2}v/ ¶ y^{2}= exp(x) sin y + (

**-**exp (x) sin y) = 0
So
the function is also harmonic for v.

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