Complex Analysis/Complex Functions/Complex Functions

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We can extend the notion of a function to complex numbers. A complex function is one that takes complex values and maps them onto complex numbers, which we write as f:\mathbb{C}\to\mathbb{C}. Unless explicitly stated, whenever the term function appears, we will mean a complex function. A function can also be multi-valued - for example, z1 / 2 has two roots for every number. This notion will be explained in more detail in later chapters.

A plot of Abs(z²) as z ranges over the complex plane.

A complex function f(z):\mathbb{C}\to\mathbb{C} will sometimes be written in the form f(z) = f(x + iy) = u(x,y) + iv(x,y), where u and v are real-valued functions of two real variables. We can convert between this form and one expressed strictly in terms of z through the use of the following identities:

x=\frac{z+\bar z}{2}, y=\frac{1}{i} \frac{z-\bar z}{2}

While real functions can be graphed on the x-y plane, complex functions map from a two-dimensional to a two-dimensional space, so visualizing it would require four dimensions. Since this is impossible we will often use the three-dimensional plots of \Re z, \Im z, and |z| to gain an understanding of what the function "looks" like.

For an example of this, take the function f(z) = z2 = (x2 − y2) + i(2xy). The plot of the surface | z2 | = x2 + y2 is shown to the right.

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