Principle

The theory in the Complex Step Differentiation is to use the relationship:

\[f(x+i\epsilon)=f(x)+i\epsilon f'(x)+O(\epsilon^2)\]

We can get the derivative:

\[f'(x)=imag(\frac{f(x+i\epsilon)}{\epsilon})\]

See Professor Nick Higham's talk on Juliacon 2018.

If you are also interested in Complex Step Differentiation, please see the relating papers:

• Computation of higher-order derivatives using the multi-complex step method

• Using Multicomplex Variables for Automatic Computation of High-Order