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real valued function from complex

• To: mathgroup at smc.vnet.net
• Subject: [mg37354] real valued function from complex
• From: strgh at mimosa.csv.warwick.ac.uk ()
• Date: Fri, 25 Oct 2002 02:46:35 -0400 (EDT)
• Organization: University of Warwick, UK
• Sender: owner-wri-mathgroup at wolfram.com

I want to define a real-valued function f[t_] from the
values of a complex-valued function on a line parametrised
by t, and then be able to handle f like any other real
function (differentiate it etc.)

A cute example is:

  Clear[rz, drz];
  rz[t_] := Re[Zeta[1/2 + I*t]];
  drz[t_] := D[rz[t], t]     (* the sort of thing I want to do *) 

so that

  Plot[{drz[t], Im[Zeta[1/2 + I*t]]}, {t, 0, 40}, 
    PlotStyle -> {RGBColor[1, 0, 0], RGBColor[0, 0, 1]}]

will work (it doesn't).
I can get a quick & dirty numerical approximation in this case
(including, as a reality check, the original function I'm 
differentiating) using something like

  Clear[rz, iz, rztable, plotzeta];
  rz[t_] := Re[Zeta[1/2 + I*t]];
  iz[t_] := Im[Zeta[1/2 + I*t]];
  rztable[tmin_, tmax_] := 
    Table[{t, rz[t]}, {t, tmin, tmax, (tmax - tmin)/50}];
  plotzeta[tmin_, tmax_] := Module[{rzapprox},
    rzapprox = Interpolation[rztable[tmin, tmax]];
    Plot[{rzapprox'[t], rz[t], iz[t]}, {t, 0, 40}, 
      PlotStyle -> {RGBColor[1, 0, 0], RGBColor[0, 1, 0],
                    RGBColor[0, 0, 1]}]
    ]

  plotzeta[0, 40]

However I'd prefer to leave the numerical approximations
till the last minute (i.e. plotting), and the interpolation 
table would need tweaking on a case-by-case basis.
Any other suggestions?  (sorry if there is an "obvious" answer).
	-- Ewart Shaw
-- 
J.E.H.Shaw   [Ewart Shaw]        strgh at uk.ac.warwick     TEL: +44 2476 523069
  Department of Statistics,  University of Warwick,  Coventry CV4 7AL,  U.K.
  http://www.warwick.ac.uk/statsdept/Staff/JEHS/
3  ((4&({*.(=+/))++/=3:)@([:,/0&,^:(i.3)@|:"2^:2))&.>@]^:(i.@[)  <#:3 6 2

• Follow-Ups:
  • Re: real valued function from complex
    • From: Daniel Lichtblau <danl@wolfram.com>