SFunction

Given the function $f(t)$ as

$$ f(t) = \frac{{(1+t^2)^2-1}}{{(1+t^2)^2+1}} $$

where $t$ is a complex number $t = x + iy$. This function can be represented as $f(t) = a(t) + i\cdot b(t)$, where $a(t)$ and $b(t)$ are the real and imaginary parts of $f(t)$ respectively.

The real part of the function $a(t)$ is derived as:

$$ a(t) = \frac{{x^4 - 6x^2y^2 + y^4 + 2x^2 - 2y^2 + 2}}{{x^4 + 2x^2y^2 + y^4 + 2x^2 + 2y^2 + 2}} $$

And the imaginary part of the function $b(t)$ is:

$$ b(t) = \frac{{4x^3y - 4xy^3}}{{x^4 + 2x^2y^2 + y^4 + 2x^2 + 2y^2 + 2}} $$

Both $a(t)$ and $b(t)$ are real-valued for any complex number $t$. The operations on $x$ and $y$ (addition, subtraction, multiplication, division, and exponentiation) yield real numbers, so both $a(t)$ and $b(t)$ are real-valued functions.

The more general form of this is known as partial fraction decomposition