Math 130 (Notes/Function-image)

# Images of Complex Functions: Trig Functions

These are some of the images that we saw in class today.

The first set shows a domain and its image as mapped by sin(z) and cos(z). Here, the domain runs from -p to p in the real part and 0 to 2 in the imaginary part. For a fixed x, the numbers z = x + iy are mapped to an hyperbola, For a fixed y, they map to ellipses. These curves are everywhere perpendicular to each other.

Extending the patch in the real direction, causes the image to continue to wrap around, over and over again. Extending the image values to higher positive values causes the ellipses to expand, eventually covering all the complex plane. Conversely, moving into the negative imaginary parts causes the image to move through the segment [-1,1] on the real axis and wraps around the plane from below.

The difference between the sine and cosine is a phase shift of 90 degrees. (You can see this by the coloring of the two images.)

The next set of diagrams show the result of mapping a patch via tan(z). The image is formed by two families of circles, one set centered on the real axis, and one on the imaginary axis. The latter shrink down to a point at i and -i.

 Math 130 (Spring 2002) web pages Created: 26 Apr 2002 Last modified: Apr 27, 2002 8:50:43 AM Comments to: `dpvc@union.edu`