# Graphs of Orthogonality

I’ve made a few more graphs to further illustrate the orthogonality of the contour maps. To recap, these contours represent the family of contours for which either the real or the imaginary part of a function of $z$ is constant. The pale gold contours represent the curves for constant real part of $F(z)$, and the red-brown the constant imaginary part.

Once again, here is a small part of the map generated by $F(z)=\cos z$:

Next is the graph generated by $F(z)=e^{z}$. It is interesting to note that the real and imaginary parts of the function are equal for $y=\frac{\pi}{4}.$

The next graph is generated by $F(z)=\ln z$. That it is simple is a consequence of the  fact that complex numbers can be represented in polar form.

The following graph is generated by $F(z)=\sqrt{z}$. The two contour plots are exact reflections of one another in the $y$ axis.

The final graph is derived from the function $F(z)=z^2$. Polynomials are perhaps the easiest category of function to rearrange into Cartesian form.

The two families of contours are different types of hyperbolae. Note that there is one contour which is not behaving itself: $y=0$. This may be because for $y=0$, $F(z)$ is no longer differentiable with respect to $z$.