MathAction SandBoxWirtinger

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        SandBoxWirtinger <-- You are here.

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

fricas

(1) -> )lib FSPECX

   >> System error:
   The value
  37981
is not of type
  LIST

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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)

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

(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)

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

(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)

$\label{eq9}{z \ {{conjugate^{\prime}}\left({z}\right)}}+{\overline z}$

(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)

$\label{eq11}\frac{{x \ {{conjugate^{\prime}}\left({x}\right)}}+{\overline x}}{2 \ {\sqrt{x \ {\overline x}}}}$

(11)

Type: Expression(Integer)

fricas

diff(%,x)

$\label{eq12}\frac{{2 \ {{x}^{2}}\ {\overline x}\ {{conjugate^{\prime \prime}}\left({x}\right)}}-{{{x}^{2}}\ {{{conjugate^{\prime}}\left({x}\right)}^{2}}}+{2 \ x \ {\overline x}\ {{conjugate^{\prime}}\left({x}\right)}}-{{\overline x}^{2}}}{4 \ x \ {\overline x}\ {\sqrt{x \ {\overline x}}}}$

(12)

Type: Expression(Integer)

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diff(%,x)

$\label{eq13}\frac{{4 \ {{x}^{3}}\ {{\overline x}^{2}}\ {{conjugate^{\prime \prime \prime}}\left({x}\right)}}+{{\left(-{6 \ {{x}^{3}}\ {\overline x}\ {{conjugate^{\prime}}\left({x}\right)}}+{6 \ {{x}^{2}}\ {{\overline x}^{2}}}\right)}\ {{conjugate^{\prime \prime}}\left({x}\right)}}+{3 \ {{x}^{3}}\ {{{conjugate^{\prime}}\left({x}\right)}^{3}}}-{3 \ {{x}^{2}}\ {\overline x}\ {{{conjugate^{\prime}}\left({x}\right)}^{2}}}-{3 \ x \ {{\overline x}^{2}}\ {{conjugate^{\prime}}\left({x}\right)}}+{3 \ {{\overline x}^{3}}}}{8 \ {{x}^{2}}\ {{\overline x}^{2}}\ {\sqrt{x \ {\overline x}}}}$

(13)

Type: Expression(Integer)

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

$\label{eq14}0$

(14)

Type: Expression(Integer)

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

$\label{eq15}0$

(15)

Type: Expression(Integer)

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

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

(16)

Type: Expression(Integer)

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

   >> Error detected within library code:
   "failed" of mode Union(Symbol,"failed") cannot be coerced to mode Symbol

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)

$\label{eq17} \mbox{\rm true}$

(17)

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))

$\label{eq18} \mbox{\rm true}$

(18)

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))

$\label{eq19}\frac{-{i \ {\overline b}}+{\overline a}+{i \ b}+ a}{2}$

(19)

Type: Expression(Complex(Integer))

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

$\label{eq20}a$

(20)

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))

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

(21)

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))

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

(22)

Type: Expression(Complex(Integer))

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

$\label{eq23}b$

(23)

Type: Expression(Complex(Integer))

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diff(real(x),x)

$\label{eq24}\frac{{{conjugate^{\prime}}\left({x}\right)}+ 1}{2}$

(24)

Type: Expression(Integer)

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diff(real(x),conjugate x)

$\label{eq25}\frac{1}{2}$

(25)

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))

$\label{eq26}\frac{{i \ {{conjugate^{\prime}}\left({x}\right)}}- i}{2}$

(26)

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))

$\label{eq27}\frac{i}{2}$

(27)

Type: Expression(Complex(Integer))

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diff(abs(x)^2,x)

$\label{eq28}\frac{2 \ {{abs \left({x}\right)}^{2}}}{x}$

(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)

$\label{eq30}c \ {{f^{\prime}}\left({x}\right)}$

(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)

$\label{eq32}{{f_{, 1}}\left({x , \:{\overline x}}\right)}+{{{f_{, 2}}\left({x , \:{\overline x}}\right)}\ {{conjugate^{\prime}}\left({x}\right)}}$

(32)

Type: Expression(Integer)

fricas

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))

   >> Error detected within library code:
   "failed" of mode Union(Symbol,"failed") cannot be coerced to mode Symbol

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))

$\label{eq34}\frac{{{\overline z}^{3}}+{{\left(-{3 \ z}+ 2 \right)}\ {{\overline z}^{2}}}+{{\left({3 \ {{z}^{2}}}+{4 \ z}\right)}\ {\overline z}}-{{z}^{3}}+{2 \ {{z}^{2}}}}{8}$

(34)

Type: Expression(Complex(Integer))

fricas

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

fricas

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

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

(35)

Type: Boolean

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

$\label{eq36}\begin{array}{@{}l} \displaystyle {\frac{{3 \ {{\overline z}^{2}}}+{{\left(-{6 \ z}+ 4 \right)}\ {\overline z}}+{3 \ {{z}^{2}}}+{4 \ z}}{8}}= \ \ \displaystyle {\frac{-{3 \ {{\overline z}^{2}}}+{{\left({6 \ z}+ 4 \right)}\ {\overline z}}-{3 \ {{z}^{2}}}+{4 \ z}}{8}}$

(36)

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)

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

(37)

Type: Boolean

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

$\label{eq38}2 \ {\log \left({\sqrt{z \ {\overline z}}}\right)}$

(38)

Type: Expression(Integer)

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diff(t2,z)

$\label{eq39}\frac{{z \ {{conjugate^{\prime}}\left({z}\right)}}+{\overline z}}{z \ {\overline z}}$

(39)

Type: Expression(Integer)

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diff(t2,conjugate z)

$\label{eq40}\frac{1}{\overline z}$

(40)

Type: Expression(Integer)

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

$\label{eq41}\frac{{x \ {{conjugate^{\prime}}\left({x}\right)}}-{\overline x}+ x}{2 \ x \ {\sqrt{x \ {\overline x}}}}$

(41)

Type: Expression(Integer)

fricas

eval(%,conjugate(x)=x)

$\label{eq42}\frac{{conjugate^{\prime}}\left({x}\right)}{2 \ {\sqrt{{x}^{2}}}}$

(42)

Type: Expression(Integer)