# Plotting a several variable complex valued function

Consider A function $f:\mathbf{C}^2\rightarrow \mathbf{C}$ defined as $$f_{\alpha, \beta}(z,w)=\frac{\alpha}{z}+\frac{\beta}{w}$$ where $\alpha$ and $\beta$ both are complex number.

I want to plot this function for different parameter value $\alpha$ and $\beta$ the function $f_{\alpha, \beta}$.

Can anyone help me in coding the same in Mathematica?

• That would require a 6D plot. For C -> C you could use 2D to 2D mappings. For this case I don't think you could do anything useful. – Sjoerd C. de Vries Sep 8 '14 at 5:48

As has been commented this could not be visualized. However, I post this as a way perhaps to explore function. In this case I have assumed the parameters are real. However, 2D sliders could be used for complex parameters.

f[a_, b_, z_, w_] := a/z + b/w
Manipulate[Column[{
ParametricPlot[{x, y}, {x, -4, 4}, {y, -4, 4}, Mesh -> 20,
ImageSize -> 400],
ParametricPlot[{Re@f[a, b, x + I  y, Complex @@ w],
Im@f[a, b, x + I  y, Complex @@ w]}, {x, -10, 10}, {y, -10, 10},
Mesh -> 50,
Epilog -> {Blue, PointSize[0.04],
Point[Through[{Re, Im}[f[a, b, Complex @@ w, Complex @@ p]]]]},
ImageSize -> 400]
}],
{{w, {1, 1}}, Locator, Appearance -> Style[\[FilledSquare], Red]},
{{p, {-0.4, -0.4}}, Locator,
Appearance -> Style[\[LightBulb], Purple]},
{a, 0, 1}, {b, 0, 1}]