Having spent several blogs on basic complex algebra and roots, we’re now in a position to examine

*complex functions*. Basically these are analogous to regular (real) functions, e.g.
f(x) = 3x – 4

f(x) = 2x

^{2}- 3x + 1
except that complex variables of the form z = x + iy are incorporated. All the basic operations we saw that apply to complex algebra, including polar forms, including r = exp[(i q)] apply also to complex functions. Note that since complex functions are dependent on the complex variable z, we typically write them as f(z).

A good way to get started is by applying simple operations to functions.

For example, let f(z) = e

^{(-3z)}
Find the real and imaginary parts of the function f(z)

Since z = x + iy, we may write:

f(z) = exp[-3(x + iy)] = exp(-3x) [exp(-i3y)]

and

f(z) = exp(-3x)[cos (3y) – isin(3y)]

Then : Re f(z) = Re exp(-3z) = exp(-3x)cos (3y)

And : Im f(z) = Im exp(-3z) = -iexp(-3x) sin (3y)

Finding numerical values for functions in many ways resembles the way we do it for real functions, simply substitute the f-value to be found into the function f(z), viz.

Find f(2i) for f(z) = - 3z

^{2}
è f(2i) = -3(2i)

^{2}= -3 (-4) = 12
More examples:

1. Find: f(-3i) for f(z) = (z + 2 – 3i) ¸ (z + 4 – i)

Again: f(-3i) = {(-3i +2 -3i)/ (-3i + 4 – i)} = (2 – 6i)/ (4 – 4i) = 1 – ½ i

2. Find f(2i -3) for f(z) = (z + 3)

^{2}(z – 5i)^{2}
è f(2i -3) = {(2i -3)+3}

^{2}(2i – 3 – 5i)^{2 }= {(-4)(18i)} = -72i
3. Let f(z) = ln r + i(q) where r = êz ê and q = Arg(z)

Find f(1):

f(1) = ln 1 + i Arg(1) but we know that q = p/4 for Arg (1)

Then:

f(1) = ln 1 + i(p/4)

4. Find: f(i p/4) for f(z) = exp(x) cos(y) + i(exp(x)sin(y)

Here, let z = r exp(i q) then q = p/4

And exp(i p/4)= cos(p/4) + isin(p/4) with r = 1

Therefore:

f (z) = exp(1)cos(p/4) +i(exp(1)sin(p/4)

f(z) = exp[(cos(p/4) +i sin(p/4)] = exp{Ö2/2 + iÖ2/2}

f(z) = 1.922 + 1.922i

Problems for the Math Maven:

- Find f(1+i) for: f(z) = 1/ (z
^{2}+ 1)

2. Find: f(-2) for f(z) = ln r + i(q), where r = êz ê and q = Arg(z)

3. Solve: (z + 1)

^{3}= z

^{3}

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