PartialFractionDecomposition

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 $v(x, y)$.