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How can complex functions be separated into real and imaginary parts?

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A graph of the function f(x) = x cubed minus 3x plus 2 is shown on a blue background, with a red curve passing through key points. Local maximum and minimum points are marked with black dots at x equals negative 1 and x equals 1, illustrating important features of the curve. — function Plot of the cubic equation f(x) = x³ − 3x + 2. The plotted points are where changes in curvature occur.

To separate a complex function into its real and imaginary parts, you represent the complex variable $z$ in the form $z = x + i y$ , where $x$ and $y$ are real variables and $i$ is the imaginary unit ( $\sqrt{- 1}$ ).123

Given a complex-valued function $f (z)$ , you can express it as:

$f (z) = f (x + i y) = P (x, y) + i Q (x, y)$

In this expression:
* $P (x, y)$ represents the real part of the function.
* $Q (x, y)$ represents the imaginary part of the function, which is the coefficient of the imaginary unit $i$ .23

Both $P$ and $Q$ are real-valued functions of the two real variables $x$ and $y$ . This decomposition is fundamental in complex analysis, particularly when examining the properties of analytic functions, where the real and imaginary components must satisfy specific constraints, such as those found in the study of fluid flow.3