The Complex Exponential Function

Consider the exponential function w = e^z given by e^{(x + yi)} = e^xcosy + e^xsiny i. The view on the left below reprents the real-part of the graph of the this function, and on the right is the imaginary part.

Notice that the function has cosine curves in the planes parallel to the yu plane (the green and blue axes) and sine curves in the yv plane (the green and white axes); these are due to the cosine and sine in the real and imaginary parts of the formula above. In both cases, there is exponential behavior in the xu and xv planes; this comes from the e^x term in each part of the function above. Toward the left, the exponential function is damping out the sine and cosine waves, and toward the right, exagerating them.

The movie shows the rotation that takes you between the two views, and the function seems to wave along. This occurs as we change from the cosine in the real part to the sine in the imaginary part, effectively making a phase shift in the wave.