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# What's a Complex Derivative?

Posted 11 years ago
 Needs["NumericalCalculus`"]ND[x, {x, -(1/2)}, 1, Method -> NIntegrate]0.752253 + 3.04624*10^-17 IND[x^4, {x, I}, 1, Method -> NIntegrate]0.0632028 + 1.11425 IAccording 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.
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Posted 11 years ago
 I would think so, at least based on these papers (link).
Posted 11 years ago
 Thanks, Ilian.  I've seen applications for fractional derivatives.  Are there any applications for complex derivatives?
Posted 11 years ago
 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]/PiIn[3]:= N[%]Out[3]= 0.0632028 + 1.11425 I
Posted 11 years ago
 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.