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Editor: Bill Page Time: 2015/03/17 21:28:28 GMT+0
Note: refs

added:

- "An Introduction to Complex Differentials and Complex Differentiability":https://mediatum.ub.tum.de/doc/631019/631019.pdf
  by Raphael Hunger

Wirtinger Derivatives or Wirtinger Calculus (also called $\mathbb{CR}$ -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

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)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

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

Type: Void

Tests

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test(diff(z,z)=1)

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

$\label{eq1} \mbox{\rm true}$

(1)

Type: Boolean

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

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

$\label{eq2} \mbox{\rm true}$

(2)

Type: Boolean

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test(diff(z,conjugate z)=0)

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

$\label{eq3} \mbox{\rm true}$

(3)

Type: Boolean

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test(diff(conjugate z,z)=0)

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

$\label{eq4} \mbox{\rm true}$

(4)

Type: Boolean

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diff(sin(x),x)

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

$\label{eq5}\cos \left({x}\right)$

(5)

Type: Expression(Integer)

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diff(sin(x),conjugate x)

$\label{eq6}0$

(6)

Type: Expression(Integer)

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diff(sqrt(x^2),x)

$\label{eq7}x \over{\sqrt{{x}^{2}}}$

(7)

Type: Expression(Integer)

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diff(sqrt(x^2),conjugate x)

$\label{eq8}0$

(8)

Type: Expression(Integer)

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diff(z*conjugate(z),z)

$\label{eq9}\overline z$

(9)

Type: Expression(Integer)

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diff(z*conjugate(z),conjugate z)

$\label{eq10}z$

(10)

Type: Expression(Integer)

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diff(sqrt(x*conjugate(x)),x)

$\label{eq11}{\overline x}\over{2 \ {\sqrt{x \ {\overline x}}}}$

(11)

Type: Expression(Integer)

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diff(%,x)

$\label{eq12}-{{\overline x}\over{4 \ x \ {\sqrt{x \ {\overline x}}}}}$

(12)

Type: Expression(Integer)

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diff(%,x)

$\label{eq13}{3 \ {\overline x}}\over{8 \ {{x}^{2}}\ {\sqrt{x \ {\overline x}}}}$

(13)

Type: Expression(Integer)

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diff(diff(diff(abs(x),x),x),x)

$\label{eq14}{3 \ {{\overline x}^{3}}}\over{8 \ {{abs \left({x}\right)}^{5}}}$

(14)

Type: Expression(Integer)

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_rule(abs(x),sqrt(x*conjugate(x))) %

$\label{eq15}{3 \ {\overline x}}\over{8 \ {{x}^{2}}\ {\sqrt{x \ {\overline x}}}}$

(15)

Type: Expression(Integer)

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diff(sqrt(x*conjugate(x)),conjugate x)

$\label{eq16}x \over{2 \ {\sqrt{x \ {\overline x}}}}$

(16)

Type: Expression(Integer)

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conjugate(diff(conjugate(sqrt(x*conjugate(x))),x))

$\label{eq17}{\sqrt{x \ {\overline x}}}\over{2 \ {\overline x}}$

(17)

Type: Expression(Integer)

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diff(%,conjugate x)

$\label{eq18}-{x \over{4 \ {\overline x}\ {\sqrt{x \ {\overline x}}}}}$

(18)

Type: Expression(Integer)

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diff(%,conjugate x)

$\label{eq19}{3 \ x}\over{8 \ {{\overline x}^{2}}\ {\sqrt{x \ {\overline x}}}}$

(19)

Type: Expression(Integer)

Non-holomorphic functions

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

Type: Void

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test(real real x = real x)

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

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

$\label{eq20} \mbox{\rm true}$

(20)

Type: Boolean

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imag(x) == -%i*(x-conjugate(x))/2

Type: Void

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test(imag imag x = 0)

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

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Compiling function imag with type Expression(Complex(Integer)) -> 
      Expression(Complex(Integer))

$\label{eq21} \mbox{\rm true}$

(21)

Type: Boolean

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real(a+%i*b)

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Compiling function real with type Polynomial(Complex(Integer)) -> 
      Expression(Complex(Integer))

$\label{eq22}{-{i \ {\overline b}}+{\overline a}+{i \ b}+ a}\over 2$

(22)

Type: Expression(Complex(Integer))

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eval(%,[conjugate(a)=a,conjugate(b)=b])

$\label{eq23}a$

(23)

Type: Expression(Complex(Integer))

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test(real(real(a) + %i*real(b)) = real(a))

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

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

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Compiling function real with type Expression(Complex(Integer)) -> 
      Expression(Complex(Integer))

$\label{eq24} \mbox{\rm true}$

(24)

Type: Boolean

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imag(a+%i*b)

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Compiling function imag with type Polynomial(Complex(Integer)) -> 
      Expression(Complex(Integer))

$\label{eq25}{{\overline b}+{i \ {\overline a}}+ b -{i \ a}}\over 2$

(25)

Type: Expression(Complex(Integer))

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eval(%,[conjugate(a)=a,conjugate(b)=b])

$\label{eq26}b$

(26)

Type: Expression(Complex(Integer))

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diff(real(x),x)

$\label{eq27}1 \over 2$

(27)

Type: Expression(Integer)

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diff(real(x),conjugate x)

$\label{eq28}1 \over 2$

(28)

Type: Expression(Integer)

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diff(imag(x),x)

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

$\label{eq29}-{i \over 2}$

(29)

Type: Expression(Complex(Integer))

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diff(imag(x),conjugate x)

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

$\label{eq30}i \over 2$

(30)

Type: Expression(Complex(Integer))

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diff(abs(x)^2,x)

$\label{eq31}\overline x$

(31)

Type: Expression(Integer)

Functions are assumed holomorphic

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f:=operator 'f

$\label{eq32}f$

(32)

Type: BasicOperator?

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diff(c*f(x),x)

$\label{eq33}c \ {{f_{\ }^{,}}\left({x}\right)}$

(33)

Type: Expression(Integer)

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

$\label{eq34}0$

(34)

Type: Expression(Integer)

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diff(f(x,conjugate x),x)

$\label{eq35}{f_{, 1}}\left({x , \:{\overline x}}\right)$

(35)

Type: Expression(Integer)

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diff(f(x,conjugate x),conjugate x)

$\label{eq36}{f_{, 2}}\left({x , \:{\overline x}}\right)$

(36)

Type: Expression(Integer)

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

$\label{eq37} \mbox{\rm false}$

(37)

Type: Boolean

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

$\label{eq38} \mbox{\rm false}$

(38)

Type: Boolean

Examples

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

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

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

$\label{eq39}{{{\overline z}^{3}}+{{\left(-{3 \ z}+ 2 \right)}\ {{\overline z}^{2}}}+{{\left({3 \ {{z}^{2}}}+{4 \ z}\right)}\ {\overline z}}-{{z}^{3}}+{2 \ {{z}^{2}}}}\over 8$

(39)

Type: Expression(Complex(Integer))

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test(diff(t1,z) = 1/2*z+1/2*conjugate(z)-3/8*(z-conjugate(z))^2)

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

$\label{eq40} \mbox{\rm true}$

(40)

Type: Boolean

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(diff(t1,conjugate z) = 1/2*z+1/2*conjugate(z)-3/8*(z-conjugate(z))^2)

$\label{eq41}\begin{array}{@{}l} \displaystyle {{{3 \ {{\overline z}^{2}}}+{{\left(-{6 \ z}+ 4 \right)}\ {\overline z}}+{3 \ {{z}^{2}}}+{4 \ z}}\over 8}= \ \ \displaystyle {{-{3 \ {{\overline z}^{2}}}+{{\left({6 \ z}+ 4 \right)}\ {\overline z}}-{3 \ {{z}^{2}}}+{4 \ z}}\over 8}$

(41)

Type: Equation(Expression(Complex(Integer)))

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test(diff(t1,conjugate z) = 1/2*z+1/2*conjugate(z)-3/8*(z-conjugate(z))^2)

$\label{eq42} \mbox{\rm false}$

(42)

Type: Boolean

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t2:=2*log(sqrt(z*conjugate(z)))

$\label{eq43}2 \ {\log \left({\sqrt{z \ {\overline z}}}\right)}$

(43)

Type: Expression(Integer)

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diff(t2,z)

$\label{eq44}1 \over z$

(44)

Type: Expression(Integer)

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diff(t2,conjugate z)

$\label{eq45}1 \over{\overline z}$

(45)

Type: Expression(Integer)

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diff(sqrt(x*conjugate(x))/x,x)+diff(sqrt(x*conjugate(x))/x,conjugate x)

$\label{eq46}{-{\overline x}+ x}\over{2 \ x \ {\sqrt{x \ {\overline x}}}}$

(46)

Type: Expression(Integer)

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eval(%,conjugate(x)=x)

$\label{eq47}0$

(47)

Type: Expression(Integer)