Compare Functions in Higher Dimensions

AsymptoticEqual, AsymptoticGreater and the other asymptotic relations are very general and can compare—in addition to functions of a single integer or real variable—functions of a complex variables, as well as multivariate functions. The comparison can be done along rays, lines or all directions at once. This example illustrates the functionality, using functions in the complex plane.

Define two functions of a complex variable.

Compare the two functions, approaching the comparison point along different rays.

Create a visualization of the ratio of the function in the complex plane centered at , with the various rays highlighted.

Since in both real directions—indeed, in the entire upper half plane—then in the real direction.

However, because in the lower half-plane, then in the complex plane.