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SandBoxWirtinger

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Editor: Bill page Time: 2014/09/14 03:15:05 GMT+0
Note:

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

Refs.

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

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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:Expression Integer,z:Kernel Expression Integer):Expression Integer == eval(D(eval(ex,[z],[%conjugate]),%conjugate),[%conjugate],[z])

   Function declaration diff : (Expression(Integer),Kernel(Expression(
      Integer))) -> Expression(Integer) has been added to workspace.

Type: Void

Tests

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test(diff(z,z)=1)

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

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

(1)

Type: Boolean

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

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

(2)

Type: Boolean

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

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

(3)

Type: Boolean

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

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

(4)

Type: Boolean

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diff(sin(x),x)

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

(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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diff(%,conjugate x)

$\label{eq17}-{x \over{4 \ {\overline x}\ {\sqrt{x \ {\overline x}}}}}$

(17)

Type: Expression(Integer)

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diff(%,conjugate x)

$\label{eq18}{3 \ x}\over{8 \ {{\overline x}^{2}}\ {\sqrt{x \ {\overline x}}}}$

(18)

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)

$\label{eq19} \mbox{\rm true}$

(19)

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))

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

(20)

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))

$\label{eq21}{-{i \ {\overline b}}+{\overline a}+{i \ b}+ a}\over 2$

(21)

Type: Expression(Complex(Integer))

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

$\label{eq22}a$

(22)

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))

$\label{eq23} \mbox{\rm true}$

(23)

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))

$\label{eq24}{{\overline b}+{i \ {\overline a}}+ b -{i \ a}}\over 2$

(24)

Type: Expression(Complex(Integer))

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

$\label{eq25}b$

(25)

Type: Expression(Complex(Integer))

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diff(real(x),x)

$\label{eq26}1 \over 2$

(26)

Type: Expression(Integer)

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diff(real(x),conjugate x)

$\label{eq27}1 \over 2$

(27)

Type: Expression(Integer)

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diff(imag(x),x)

   Conversion failed in the compiled user function diff .

   Cannot convert from type Expression(Complex(Integer)) to Expression(
      Integer) for value
     _
   %ix - %i x
   ----------
        2

diff(imag(x),conjugate x)

   Conversion failed in the compiled user function diff .

   Cannot convert from type Expression(Complex(Integer)) to Expression(
      Integer) for value
     _
   %ix - %i x
   ----------
        2

diff(abs(x)^2,x)

$\label{eq28}\overline x$

(28)

Type: Expression(Integer)

Functions are assumed holomorphic

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f:=operator 'f

$\label{eq29}f$

(29)

Type: BasicOperator?

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diff(c*f(x),x)

$\label{eq30}c \ {{f_{\ }^{,}}\left({x}\right)}$

(30)

Type: Expression(Integer)

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

$\label{eq31}0$

(31)

Type: Expression(Integer)

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

$\label{eq32}{f_{, 1}}\left({x , \:{\overline x}}\right)$

(32)

Type: Expression(Integer)

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

$\label{eq33}{f_{, 2}}\left({x , \:{\overline x}}\right)$

(33)

Type: Expression(Integer)

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

$\label{eq34} \mbox{\rm true}$

(34)

Type: Boolean

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

$\label{eq35} \mbox{\rm true}$

(35)

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))

$\label{eq36}{{{\overline z}^{3}}+{{\left(-{3 \ z}+ 2 \right)}\ {{\overline z}^{2}}}+{{\left({3 \ {{z}^{2}}}+{4 \ z}\right)}\ {\overline z}}-{{z}^{3}}+{2 \ {{z}^{2}}}}\over 8$

(36)

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)

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

(37)

Type: Boolean

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

$\label{eq38}\begin{array}{@{}l} \displaystyle {{{3 \ {{\overline z}^{2}}}+{{\left(-{6 \ z}+ 4 \right)}\ {\overline z}}+{3 \ {{z}^{2}}}+{4 \ z}}\over 8}= \ \ \displaystyle {{-{3 \ {{\overline z}^{2}}}+{{\left({6 \ z}+ 4 \right)}\ {\overline z}}-{3 \ {{z}^{2}}}+{4 \ z}}\over 8}$

(38)

Type: Equation(Expression(Integer))

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

$\label{eq39} \mbox{\rm false}$

(39)

Type: Boolean

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

$\label{eq40}2 \ {\log \left({\sqrt{z \ {\overline z}}}\right)}$

(40)

Type: Expression(Integer)

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diff(t2,z)

$\label{eq41}1 \over z$

(41)

Type: Expression(Integer)

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diff(t2,conjugate z)

$\label{eq42}1 \over{\overline z}$

(42)

Type: Expression(Integer)