Additional Problems II

 

1.     Where in the complex plane are the following functions analytic? If the derivative exists in a domain, find an expression for f’(z) in terms of z or x and y.

 

(a)   -xy+(i/2)(x²-y²)

 

(b)  z²/(e^xcosy+ie^xsiny)

 

2.     Where in the complex plane is the following function analytic? (DO not consider the origin)

 

(a)     rcosq+ir

 

3.     For what values of complex variable z do the following functions have derivatives?

 

(a)   z^-5

 

(b)  x

 

(c)   z⁵+z

 

(d)  f(z)=|z|², |z|≤1, but f(z)=z², |z|>1

 

(e)   f(z)=z, |z|≥1, but f(z)=1, |z|<1