Mumbai University > Information Technology, Computer Engineering > Sem 3 > Applied Mathematics 3

Marks: 6M

Year: Dec 2015

Let f(z)=u+i v be analytic

We have real part,

$u = x^3 - 3x^2y - y^3$

Differentiating with respect to

Therefore, $v_x=3x^2-6xy$

Again, differentiating with respect to x, we get

$v_{xx} = 6x - 6y$

Now, differentiating u with respect to y, we get

$u_y=0-3x^2-3y^2$

Again, differentiating with respect to y, we get

$u_{yy} = -6y\\ u_{xx} + u_{yy} = 6x - 6y - 6y$

Therefore, u does not satisfy Laplace equation and hence it is not a harmonic function.

Thus, v is not analytic function

f(z)= u + i v

Hence, the statement is true. There does not exist an analytic function whose real part is

$x^3-3x^2 y-y^3$