Wirtinger Derivatives or Wirtinger Calculus (also called -Calculus)

Reference

• http://en.wikipedia.org/wiki/Wirtinger_derivatives
• The Complex Gradient Operator and the CR-Calculus by Ken Kreutz-Delgado
• Appendix A: Wirtinger Calculus by Robert F. H. Fischer
• Wirtinger derivatives, Beltrami equation & ellipse fields by Thomas Schramm
• Non-analytic functions of a complex variable or alternate by E. R. Hedrick
• Pseudo-Conformal Geometry of Polygenic Functions of Several Complex Variables by Edward Kasner and John De Cicco
• A Complete Characterization of the Derivative of a Polygenic Function by Edward Kasner
• Polygenic Functions Of Hypercomplex Variables by P.W.Ketchum and Ted Martin
• An Introduction to Complex Differentials and Complex Differentiability by Raphael Hunger

(1) -> )lib FSPECX

   FunctionalSpecialFunction is now explicitly exposed in frame initial

   FunctionalSpecialFunction will be automatically loaded when needed 
      from /var/aw/var/LatexWiki/FSPECX.NRLIB/FSPECX

diff(ex,z) == eval(D(eval(ex,[z],[%conjugate]),%conjugate),[%conjugate],[z])

Tests

test(diff(z,z)=1)

Compiling function diff with type (Variable(z), Variable(z)) -> 
      Polynomial(Integer)

(1)

test(diff(conjugate z,conjugate z)=1)

   >> System error:
   #<SB-SYS:FD-STREAM for "file /var/aw/var/LatexWiki/FSPECX.NRLIB/FSPECX.fasl" {100265C7A3}>
is a fasl file compiled with SBCL 1.1.1, and can't be loaded into SBCL
2.2.9.debian.

Non-holomorphic functions

real(x) == (x+conjugate(x))/2

test(real real x = real x)

Compiling function real with type Variable(x) -> Expression(Integer)

   >> System error:
   #<SB-SYS:FD-STREAM for "file /var/aw/var/LatexWiki/FSPECX.NRLIB/FSPECX.fasl" {100283DBD3}>
is a fasl file compiled with SBCL 1.1.1, and can't be loaded into SBCL
2.2.9.debian.

Functions are assumed holomorphic

f:=operator 'f

(2)

diff(c*f(x),x)

Compiling function diff with type (Expression(Integer), Variable(x))
       -> Expression(Integer)

(3)

diff(c*f(x),conjugate x)

   >> System error:
   #<SB-SYS:FD-STREAM for "file /var/aw/var/LatexWiki/FSPECX.NRLIB/FSPECX.fasl" {1002CA70E3}>
is a fasl file compiled with SBCL 1.1.1, and can't be loaded into SBCL
2.2.9.debian.

Examples

t1:=real(z)^2+%i*imag(z)^3

Compiling function real with type Variable(z) -> Expression(Integer)

   >> System error:
   #<SB-SYS:FD-STREAM for "file /var/aw/var/LatexWiki/FSPECX.NRLIB/FSPECX.fasl" {1002D881F3}>
is a fasl file compiled with SBCL 1.1.1, and can't be loaded into SBCL
2.2.9.debian.