# Complex Functions as Mappings

- Author:
- Prof. Caine

This applet helps you to explore complex functions f(z)=u(x,y)+iv(x,y), where z=x+iy. You can input the real part u(x,y) and the imaginary part v(x,y) as formulas in x and y.

Input the real part u(x,y) and the imaginary part v(x,y) of the complex function f. The exponential function is the first example. Click and drag the point Z in the domain on the left and observe the corresponding W=f(Z) in the codomain on the right. By Right-clicking on the point Z, you can select Trace On to trace the point Z as it moves and similarly for W. Repeat this to turn off the trace. Click on the cycle symbol in the upper right corner of the applet to refresh the screen and clear the traces. By clicking on the blue check box, you can turn on a grid with lower left corner at Z and its image in the codomain. Adjusting the scale slider, you can enlarge or shrink the grid. Moving the point Z will move the grid and its image as well. Similarly, you can explore the image of a disk centered at Z.
Clicking the last checkbox reveals a moveable test point L and epsilon-disk centered at L for playing the epsilon-delta game. Move Z in the domain to a location and then move the test point L to a candidate value for the limit of f(z) as z approaches Z. Set epsilon using the slider and check the Show circular scale grid at Z and image. Adjust the scale to see if you can find a disk about Z whose image lands in the disk about L.