In this section, we will discuss
- Complex functions
- Polar form of complex numbers
- Limit of a function
- Differentiability
- Analytic Functions
Complex Functions
The last section developed a basic theory of complex
numbers. Now we turn our attention to the functions
of complex numbers. they are defined in a similar
way to the functions of real numbers that you studied
in calculus; the only difference is that they operate
real numbers rather than on complex numbers. This
section primarily focuses on very basic functions,
their representations and the properties associated
with them.
A complex-valued function ‘f’ of a complex variable
‘z’ is a rule that assigns to each complex number ‘z’
in a set D, a one and only one complex number ‘w’, we write w=f(z) and call w the image of z under f. A
very simple example of a complex-valued function is
given by the formula w = f(z) = `z^4`. The set D is
called the domain of f, and the set of all images is
{w = f(z): z`\in`D}
When the context is obvious, we omit the phrase
complex-valued function, and simply refer to a
function f, or to a complex function f.
Just as z can be expressed by its real and
imaginary parts, 𝑧 = 𝑥 + 𝑖𝑦, we write 𝑓(𝑧) =
𝑤 = 𝑢 + 𝑖𝑣 where u and v are the real and
imaginary parts of w, respectively. Doing so
gives us the representation
𝑤 = 𝑓(𝑧) = 𝑓(𝑥, 𝑦) = 𝑓(𝑥 + 𝑖𝑦) = 𝑢 + 𝑖𝑣
Since u and v depends on x and y, they can be
considered to be a real valued functions of the
real variables x and y, 𝑢 = 𝑢(𝑥, 𝑦) and 𝑣 = 𝑣(𝑥, 𝑦)
combining these two ideas we often write a
complex function ‘f’ in the form of
𝑓(𝑧) = 𝑓(𝑥 + 𝑖𝑦) = 𝑢(𝑥, 𝑦) + 𝑖 𝑣(𝑥, 𝑦)
we now give several examples that illustrates how
to express a complex function.
Example:1
Write f(z) = `z^4` in the form of 𝑓(𝑧) = 𝑢(𝑥, 𝑦) + 𝑖 𝑣(𝑥, 𝑦) using the binomial formula we obtain
𝑓(𝑧) = 𝑓(𝑥 + 𝑖𝑦) = `(x + iy)^4`
= `x^4 + 4 x^3 (iy) + 6 x^2 (iy)^2 + 4x(iy)^3 + (iy)^4`
= `x^4 + 4 x^3 iy - 6 x^2 y^2 - 4x i y^3 + y^4`
= `x^4 - 6 x^2 y^2 + y^4 + i(4 x^3 y - 4x y^3)`
= 𝑢(𝑥, 𝑦) + 𝑖 𝑣(𝑥, 𝑦)
Example:2
Express the function 𝑓(𝑧) = 𝑧̅ Re(z) + `z^2` + Im(z) in the form of 𝑓(𝑧) = 𝑢(𝑥, 𝑦) + 𝑖 𝑣(𝑥, 𝑦) using elementary properties of complex numbers, it
follows that
𝑓(𝑧) = `(x - iy)x + z^2 + y`
= `x^2 - ixy + x^2 + 2ixy - y^2 + y`
= `2x^2 - y^2 + y + ixy`
𝑢(𝑥, 𝑦) = `2x^2 - y^2 + y` , 𝑣(𝑥, 𝑦) = `xy`
Polar form of complex functions
Using `z = re^(i\theta)` in the expression of a complex
function f may be convenient. It gives us polar
representation `f(z) = f(re^(i\theta)) = u(r, θ) + iv(r, θ)`
where U and V are real functions of real variables
r and θ.
Analytic Functions: If the derivative 𝑓'(𝑧) exists at all points z of a region R in a complex plane, then 𝑓(𝑧) is said to be analytic in R.
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