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) = z

^{2}in the form f(z) = u(x,y) + iv(x,y)

If z = (x + iy) then: z

^{2}= (x + iy)

^{2}= x

^{2}+ i2xy - y

^{2}

= (x

^{2 }– y^{2}) + i2xyThe 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) = x

^{2}+ y^{2}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) = = 4x

^{2}+ i4y^{2} 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] = i2ySo: 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)= z

^{2}in polar form.z

^{2}= r^{2}exp(i2(q)) = r^{2}(cos (2q) + isin(2 q))Therefore:

u(r, q) = r

^{2}(cos (2q) and:v (r, q) = r

^{2}(sin (2q))*:*

__Problems for the Math Maven__1) Given u(x,y) + iv(x,y) = 2x

^{2}+ i2y^{2}find f(z,z*)2) Express f(z)= z

^{2}+ z – 3 in polar form3) Solve for x and y:

8x 2 + 3iy - 4 = 8y – 4iy

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