Functions of a Complex Variable Revisited

 Functions of a complex variable can be most generally written in the form:

f(x + iy) = u + iv

and since u,v depend on x and y, they can be considered as real functions of the real variables x and y such that:


u = u(x,y) and v = v(x,y)


Example: write f(z) = z²  in the form f(z) = u(x,y) + iv(x,y)

If z = (x + iy) then:   z² = (x + iy)² = x² + i2xy  - y²  

=  (x² – y²) + i2xy

The last step above shows how the complex function is separated into two parts, one with the factor i, the other without. The one with the factor applies to the function v(x,y) so:

v(x,y) = 2xy   

While:   u(x,y) = x² + y²

Conversely, of course, one can be given the functions u(x,y) and v(x,y) then be asked to find f(z), e.g. in terms of z and-or its complex conjugate, z*.

Example (1):

Given u(x,y) + iv(x,y) =  = 4x² + i4y²

 Find f(z,z*)

Again, let:  z = x + iy, and z* = x – iy

Adding:

z + z* = (x + iy) + (x -  iy) = 2x

So we see: x = (z + z*)/2

Now, subtracting: (z – z*) = [x + iy – x + iy] =  i2y

So: y = (z – z*)/ 2i

 

Since we have both x and y we can now formulate the function f(z,z*):

f(z,z*) = 4[(z + z*)/2]²  + i4[(z – z*)/ 2i]²

Before leaving the basics of complex functions, it’s important to note that a polar form is also used, viz.

f (z) = f(r exp(i(q)) = u(r, q) + iv(r, q)

Example (2) 

Express f(z)= z²   in polar form.

z² = r² exp(i2(q)) = r² (cos (2q) + isin(2 q))

Therefore:

u(r, q) =  r²(cos (2q)    and:

v (r, q) =    r²(sin (2q)) 

Problems for the Math Maven:

1)  Given u(x,y) + iv(x,y) = 2x² + i2y² find f(z,z*)

2)  Express f(z)= z² + z – 3 in polar form

3) Solve for x and y:   

8x 2 + 3iy  -  4    = 8y   – 4iy