Attempt any four of the following: a) Prove that the function f(z) = |z|^2 is not analytic. b) Prove that: |cosh z|^2 = sinh^2 x + cos^2 y c) Find the point at which the functio... Attempt any four of the following: a) Prove that the function f(z) = |z|^2 is not analytic. b) Prove that: |cosh z|^2 = sinh^2 x + cos^2 y c) Find the point at which the function f(z) = xy^2 is analytic. d) Express z = (1 - 2i) in polar form. e) Prove that the two power series Σ an z^n and Σ n an' z^n have the same radius of convergence. f) Find the absolute value and conjugate of z = (2 + i)(4 + 3i). Read more

Steps to Solve

1. Prove that ( f(z) = |z|^2 ) is not analytic

To show that ( f(z) = |z|^2 ) is not analytic, we can check the Cauchy-Riemann equations. Recall that ( |z|^2 = x^2 + y^2 ), where ( z = x + iy ).

The real part ( u(x, y) = x^2 + y^2 ) and the imaginary part ( v(x, y) = 0 ).

The partial derivatives are:

• ( \frac{\partial u}{\partial x} = 2x )
• ( \frac{\partial u}{\partial y} = 2y )
• ( \frac{\partial v}{\partial x} = 0 )
• ( \frac{\partial v}{\partial y} = 0 )

Now checking the Cauchy-Riemann equations:

• ( \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \implies 2x = 0 ) (only true at ( x = 0 ))
• ( \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \implies 2y = 0 ) (only true at ( y = 0 ))

Since the equations do not hold generally, ( f(z) ) is not analytic.

2. Prove that ( |\cosh z|^2 = \sinh^2 x + \cos^2 y )

We start with the definition of ( \cosh z ): $$ \cosh z = \cosh(x + iy) = \cosh x \cos y + i \sinh x \sin y $$

Calculating ( |\cosh z|^2 ): $$ |\cosh z|^2 = (\cosh x \cos y)^2 + (\sinh x \sin y)^2 $$

Expanding: $$ |\cosh z|^2 = \cosh^2 x \cos^2 y + \sinh^2 x \sin^2 y $$

Using the trigonometric identity ( \cos^2 y + \sin^2 y = 1 ): $$ |\cosh z|^2 = \sinh^2 x + (\cosh^2 x - \sinh^2 x) \cos^2 y = \sinh^2 x + \cos^2 y $$

3. Find the point where ( f(z) = xy^2 ) is analytic

To find where ( f(z) = xy^2 ) is analytic, identify the real and imaginary parts:

Let ( u(x,y) = xy^2 ) and ( v(x,y) = 0 ).

Compute the partial derivatives:

• ( \frac{\partial u}{\partial x} = y^2 )
• ( \frac{\partial u}{\partial y} = 2xy )
• ( \frac{\partial v}{\partial x} = 0 )
• ( \frac{\partial v}{\partial y} = 0 )

Now check the Cauchy-Riemann equations:

1. ( y^2 = 0 ) (implying ( y = 0 ))
2. ( 2xy = 0 ) (hold true if ( x = 0 ) or ( y = 0 ))

Thus, ( f(z) ) is analytic at points where ( y = 0 ) (on the real axis).

4. Express ( z = (1 - 2i) ) in polar form

To express ( z ) in polar form:

• Find ( r = |z| = \sqrt{(1)^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5} )
• Find the angle ( \theta = \tan^{-1}\left(\frac{-2}{1}\right) = \tan^{-1}(-2) )

Thus, in polar form, ( z = \sqrt{5} (\cos \theta + i \sin \theta) ).