PartialFractionDecomposition - crowlogic/arb4j GitHub Wiki

For rational holomorphic functions, which can be expressed as the ratio of two polynomials, it is indeed possible to split them into separate real and imaginary parts.

Let's consider a rational holomorphic function:

$$f(z) = \frac{P(z)}{Q(z)}$$

where $P(z)$ and $Q(z)$ are polynomials in the complex variable $z$. To split this function into its real and imaginary parts, we can write it as:

$$f(z) = u(x, y) + i\cdot v(x, y)$$

where $u(x, y)$ and $v(x, y)$ are real-valued functions of the real variables $x$ and $y$ corresponding to the real and imaginary parts of $f(z)$, respectively.

To find $u(x, y)$ and $v(x, y)$, we express $z$ in terms of $x$ and $y$ as $z = x + iy$. We substitute this into the function $f(z)$ and separate the real and imaginary parts by collecting the real terms and the imaginary terms.

Let $P(x, y)$ and $Q(x, y)$ represent the polynomials $P(z)$ and $Q(z)$ after substituting $z = x + iy$. Then the real and imaginary parts are given by:

$$u(x, y) = \text{Re}[f(z)] = \frac{P(x, y)}{Q(x, y)}$$

$$v(x, y) = \text{Im}[f(z)] = \frac{R(x, y)}{Q(x, y)}$$

where $R(x, y)$ represents the imaginary part of $P(z)$ after substituting $z = x + iy$.

In this way, rational holomorphic functions can be split into real and imaginary parts in terms of real-valued functions $u(x, y)$ and $v(x, y)$.