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A complex function is a function from complex numbers to complex numbers, its domain is complex z=complex(x,y).
- Real function f(x,y) in the real space (x,y,f) :

- Real function f(x) on the real axis x is cross-section in above figure on y-axis at y=0 :

- Complex function re(f) = re(f(x,y)), Real part of complex value in the real space (x,y,re(f)) :

- Complex function f(z) = re(f) + im(f) i, re(f) = re(f(x)) and im(f) = im(f(x)), Real (blue) + imaginary (red) part on the real axis x is cross-section in above figure on y-axis at y=0 :

- re(f) = re(f(complex(x,y))), Real part of complex value in the complex plane :

- re(f) = re(f(complex(x,y))), x is independent variable and y is parameter, Real part of complex value on the real axis x is cross-section on y axis in above figure.
It is the same as Figure 4 when y=0.

- re(f) = re(f(complex(x,y))), x is parameter and y is independent variable, Real part of complex value on the imaginary axis y is cross-section on x axis:

- im(f) = im(f(x,y)), Imaginary part of complex value in the real plane (x,y) :

- im(f) = im(f(x)), Imaginary part of complex value on the real axis x is cross-section in above figure on y-axis at y=0 :

- im(f) = im(f(complex(x,y))), Imaginary part of complex value on the complex plane:

- im(f) = im(f(complex(x,y))), x is independent variable and y is parameter, Imaginary part of complex value on the real axis x is cross-section on y axis in above figure.
It is the same as Figure 9 when y=0.

- im(f) = im(f(complex(x,y))), x is parameter and y is independent variable, Imaginary part of complex value on the imaginary axis y is cross-section on x axis:

- f = figure 6 + figure 10 = re(f)+ im(f) i = Real (blue) + imaginary (red) part of complex value on the real axis x is cross-section on y axis in above figure.
It is the same as Figure 4 when y=0:

- f = figure 7 + figure 12 = re(f) + im(f) i = Real (blue) + imaginary (red) part of complex value on the imaginary axis y is cross-section on x axis:

- abs(f) = abs(f(complex(x,y))), Absolute value of complex value on the complex plane:

## Reference

Function category: complex functions
Function category: complex math