What's a Complex Derivative?

Questions
,
1 
Needs["NumericalCalculus`"]

ND[x, {x, -(1/2)}, 1, Method -> NIntegrate]
0.752253 + 3.04624*10^-17 I

ND[x^4, {x, I}, 1, Method -> NIntegrate]
0.0632028 + 1.11425 I

According to the documentation for Version 9, ND can calculate fractional and complex derivatives, as shown above.
I know what fractional derivatives are but I’ve never heard of complex derivatives.

2

Complex derivative most often means derivative of a complex function. I don’t see it mentioned anywhere but complex derivatives might be defined in a way analogous to fractional derivatives: by using an imaginary exponent in Cauchy’s integral formula.

Incidentally, it is stated in the Documentation that Method → NIntegrate does not work unless the differentiated function is in analytic form.

3

There are different possible definitions, the one ND uses is in terms of Cauchy integrals. For example, the 2nd derivative of z^4 at a=1 is

In[1]:= With[{n = 2, a = 1}, n!/(2 Pi) Integrate[z = a + Exp[I t]; z^4/(z - a)^n, {t, 0, 2 Pi}]]

Out[1]= 12

In this representation, let the derivative order n be I instead of 2:

In[2]:= With[{n = I, a = 1}, n!/(2 Pi) Integrate[z = a + Exp[I t]; z^4/(z - a)^n, {t, 0, 2 Pi}]]

Out[2]= -(12/85 - (48 I)/85) I! Sinh[Pi]/Pi

In[3]:= N[%]

Out[3]= 0.0632028 + 1.11425 I
4

Thanks, Ilian.
I’ve seen applications for fractional derivatives. Are there any applications for complex derivatives?

5

I would think so, at least based on these papers (link).