Derivative of real functions including Re and Im

Question

When deriving functions using Re, Im or Arg (and probably some other functions as well), Mathematica gives nonsensical results, e.g.

Assuming[Element[x, Reals], D[Re[Gamma[I x]], x]]
(*
==> I Gamma[I x] PolyGamma[0, I x] Re'[Gamma[I x]]
*)

Of course the wanted result would be -Im[Gamma[I x] PolyGamma[0, I x]]. Now in this case, it is easy to resolve, but often those functions are deeply inside a complicated expression. Therefore my question is:

Is there any way to get reasonable derivatives for real functions of real parameters which involve Re, Im etc. of complex expressions depending on those variables?

Answer 1

The functions Re and Im (just as Conjugate) don't satisfy the Cauchy-Riemann differential equations and are therefore not analytic. That means their derivative is not uniquely defined in the complex plane. That's the reason why Re' and Im' can't be simplified.

Therefore, we have to be more specific about how we want the limit to be done that corresponds to the desired derivative. The cleanest way of doing that is this simple replacement for the derivative:

Limit[(Gamma[I (x + ε)] - Gamma[I x])/ε, ε -> 0]

$i \Gamma (i x) \psi ^{(0)}(i x)$

Now I'll try to apply this to the real part instead:

Assuming[Element[x, Reals], Limit[(Re[Gamma[I (x + ε)]] - Re[Gamma[I x]])/ε, ε -> 0]]

$-\Im(\psi ^{(0)}(i x)) \Re(\Gamma (i x))-\Im(\Gamma (i x)) \Re(\psi ^{(0)}(i x))$

So we don't get any of the pesky Re' here. For those who don't like the $\LaTeX$ format, here is the real output:

-Im[PolyGamma[0, I x]] Re[Gamma[I x]] - Im[Gamma[I x]] Re[PolyGamma[0, I x]]

Edit: generalization

Motivated by the discussion of Heike's answer, I wrote a definition of a directional derivative in an arbitrary direction in the complex plane. I did this to show that such a definition can be made without in any way modifying SystemOptions (see in particular "More Information").

dirDeriv[f_, var_, angle_: 0] := Module[{g, x},
  g = f /. var -> x;
  Assuming[x ∈ Reals,
   D[g, x] /. (h_'[p_] :> h'[p]/D[p, x]) /. (h_'[p_] :> 
       Limit[(h[p /. x -> x + ε Exp[I angle]] - 
            h[p])/ε/Exp[I angle], ε -> 0]) /. 
    x -> var
   ]
  ]

The first argument is the function to be differentiated, the second argument is the name of the independent variable, which will be treated as a real number. The third (optional) argument is the phase angle of the line in the complex plane along which the limit for the derivative is taken. By default this angle is zero corresponding to the real axis (the result depends on the angle only if the first argument is non-analytic).

Here is how to apply this function as a replacement for the standard derivative:

dirDeriv[Re[Gamma[I x]], x]

$-\Im(\psi ^{(0)}(i x)) \Re(\Gamma (i x))-\Im(\Gamma (i x)) \Re(\psi ^{(0)}(i x))$

A simpler example is

dirDeriv[Re[Exp[I x^2]], x]

$-2 x \sin \left(x^2\right)$

The function works by doing the standard derivative and then looking for any occurrences of Re', Im' and others (in fact, anything looking like h_'[p_]). These patterns are then replaced by first undoing the chain rule that Mathematica automatically applies (that's the division by D[p, x]), and then calculating the directional derivative as a limit analogous to the example above.

With this function one can easily explore the direction dependence of the derivative for non-analytic functions. For example,

Table[dirDeriv[Re[x], x, angle], {angle, 0, Pi, Pi/4}]

{1, 1/2 - I/2, 0, 1/2 + I/2, 1}

Edit 2: numerical functions

The above discussion concerns symbolic directional derivatives of possibly non-analytic functions. The situation is in some ways much simpler if the goal is to do numerical differentiation, i.e., calculate the derivative of a function f[x] which is numeric when the argument x is numeric.

For that case, one can go straight to the numerical approach by doing this:

Needs["NumericalCalculus`"];
ND[Re[Gamma[I x]], x, 1]

With the numerical derivative ND, you can for example make plots directly. Here is a comparison of the symbolic result with the numerical one:

Plot[{
  -Im[PolyGamma[0, I x0]] Re[Gamma[I x0]] - 
   Im[Gamma[I x0]] Re[PolyGamma[0, I x0]], 
   ND[Re[Gamma[I x]], x, x0]},
  {x0, 0, 5}, PlotStyle -> {Thin, Dotted}]

Answer 2

Since the derivative of Re[f[x]] and Im[f[x]] with respect to a real variable x is Re[f'[x]] and Im[f'[x]], respectively, one strategy for calculating the derivatives would be exclude Re and Im when differentiating the expression and then to replace any occurrences of D[Re[a], b] and D[Im[a], b] with Re[D[a, b]] and Im[D[a, b]].

First we need to prevent Re and Im from being differentiated. We do this by adding them to the list given by SystemOptions["DifferentiationOptions" -> "ExcludedFunctions"] using SetSystemOptions:

excl = SystemOptions["DifferentiationOptions" -> "ExcludedFunctions"][[1, 2, 1, 2]];
SetSystemOptions["DifferentiationOptions" -> 
  "ExcludedFunctions" -> Union[Join[excl, {Re, Im}]]];

Now, when you're trying to differentiate an expression involving Re and Im Mathematica will apply the chain rule but leaves expressions of the form D[Re[f[x], x] unevaluated, e.g.

D[Sin[Re[I x^2 + 5 x]], x]

The next step is to replace the derivatives of the real or imaginary part of a function with the real or imaginary part of the derivative of the function:

deriv[exp_, x__] := D[exp, x] /. {HoldPattern[D[Re[a_], b__]] :> Re[D[a, b]],
    HoldPattern[D[Im[a_], b__]] :> Im[D[a, b]]}

Example

deriv[Re[Gamma[I x]], x]

(* ==>  -Im[Gamma[I x] PolyGamma[0, I x]] *)

Answer 3

Ok, Jens' directional derivative functions convinced me that fixing after the fact is indeed possible; however it doesn't give nice results for non-specified symbolic functions. On the other hand, Heike's solution handles those quite nicely, but needs to modify global state, which I consider acceptable for this case (but not everyone does, as Jens' comments prove), but nevertheless solutions without modifying global state are IMHO generally preferrable. By combining elements from both approaches I've come to the following solution which seems to work quite well and also treats the cases of Abs and Arg:

deriv[expr_,var_]:=
  D[expr,var]//.{Re'[e_]:>Re[D[e,var]]/D[e,var],
                 Im'[e_]:>Im[D[e,var]]/D[e,var],
                 Arg'[e_]:>Im[D[e,var]/e]/D[e,var],
                 Abs'[e_]:>(Re[e] Re[D[e,var]] + Im[e] Im[D[e,var]])/     
                           (Abs[e] D[e,var])}

With this I get

deriv[Re[Gamma[I x]],x]
(*
==> -Im[Gamma[I x] PolyGamma[0, I x]]
*)
deriv[Re[f[Re[f[x]]]],x]
(*
==> Re[Re[f'[x]] f'[Re[f[x]]]]
*)
%//Simplify
(*
==> Re[f'[x]] Re[f'[Re[f[x]]]]
*)