Differntial coefficients and derivatives

Exercise3.2

1. Differentiate the following functions.

(a)

$z^3-2z^2 + 3z$

(b)

$(z^2 + i)^3$

(c)

$\frac{z-i}{z+i}$

2. Differentiate the following functions.

(a)

$\tan{z}$

(b)

$${\frac{1}{\cos{z}}}$$

(c)

$\sqrt{z^2 + 1}$

(d)

$\sin^{2}{z}$

(e)

$\log(z^2 + 4i)$

(f)

$${i^{\cos{z}}}$$

(g)

$\sin^{-1}{(z-i)}$

(h)

$\log(z + \sqrt{z^2 + 1})$

(i)

$\log(\sin^{-1}{z})$

(j)

$${z^z}$$

3. When $z = x + iy$, check the analyticity of the following functions and if it is analytic, find the derivative.

(a)

$x - iy$

(b)

$x^2 - y^2 + 2ixy$

(c)

$(x^2 - y^2 - 3x + 2) + i(2xy - 3y)$

(d)

$${\frac{x + iy}{x^2 + y^2}}$$