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SandBoxWirtinger

Edit detail for SandBoxWirtinger revision 9 of 21

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Editor: Bill Page Time: 2014/08/16 00:26:58 GMT+0
Note: CR-calculus

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

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
abs2sqrt := rule abs(a+%i*b)==sqrt(a^2+b^2)

$\label{eq1}{abs \left({{i \ b}+ a}\right)}\mbox{\rm = =}{\sqrt{{{b}^{2}}+{{a}^{2}}}}$

(1)

Type: RewriteRule?(Integer,Complex(Integer),Expression(Complex(Integer)))

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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)

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

(2)

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)

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

(3)

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)

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

(4)

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)

$\label{eq5} \mbox{\rm true}$

(5)

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)

$\label{eq6}\cos \left({x}\right)$

(6)

Type: Expression(Integer)

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diff(sin(x),conjugate x)

$\label{eq7}0$

(7)

Type: Expression(Integer)

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diff(sqrt(x^2),x)

$\label{eq8}x \over{\sqrt{{x}^{2}}}$

(8)

Type: Expression(Integer)

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

$\label{eq9}0$

(9)

Type: Expression(Integer)

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diff(z*conjugate(z),z)

$\label{eq10}\overline z$

(10)

Type: Expression(Integer)

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

$\label{eq11}z$

(11)

Type: Expression(Integer)

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

$\label{eq12}{\overline x}\over{2 \ {\sqrt{x \ {\overline x}}}}$

(12)

Type: Expression(Integer)

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

$\label{eq13}-{{\overline x}\over{4 \ x \ {\sqrt{x \ {\overline x}}}}}$

(13)

Type: Expression(Integer)

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

$\label{eq14}1 \over{4 \ {\sqrt{x \ {\overline x}}}}$

(14)

Type: Expression(Integer)

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

$\label{eq15}x \over{2 \ {\sqrt{x \ {\overline x}}}}$

(15)

Type: Expression(Integer)

Non-holomorphic functions

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

Type: Void

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

Type: Void

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real(a+%i*b)

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

$\label{eq16}{-{i \ {\overline b}}+{\overline a}+{i \ b}+ a}\over 2$

(16)

Type: Expression(Complex(Integer))

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

$\label{eq17}a$

(17)

Type: Expression(Complex(Integer))

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imag(a+%i*b)

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

$\label{eq18}{{\overline b}+{i \ {\overline a}}+ b -{i \ a}}\over 2$

(18)

Type: Expression(Complex(Integer))

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

$\label{eq19}b$

(19)

Type: Expression(Complex(Integer))

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diff(real(x),x)

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

$\label{eq20}1 \over 2$

(20)

Type: Expression(Integer)

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diff(real(x),conjugate x)

$\label{eq21}1 \over 2$

(21)

Type: Expression(Integer)

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diff(imag(x),x)

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

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

$\label{eq22}-{i \over 2}$

(22)

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))

$\label{eq23}i \over 2$

(23)

Type: Expression(Complex(Integer))

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diff(abs(x)^2,x)

$\label{eq24}\overline x$

(24)

Type: Expression(Integer)

Functions are assumed holomorphic

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f:=operator 'f

$\label{eq25}f$

(25)

Type: BasicOperator?

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diff(c*f(x),x)

$\label{eq26}c \ {{f_{\ }^{,}}\left({x}\right)}$

(26)

Type: Expression(Integer)

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

$\label{eq27}0$

(27)

Type: Expression(Integer)

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

$\label{eq28}{f_{, 1}}\left({x , \:{\overline x}}\right)$

(28)

Type: Expression(Integer)

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

$\label{eq29}{f_{, 2}}\left({x , \:{\overline x}}\right)$

(29)

Type: Expression(Integer)

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

$\label{eq30} \mbox{\rm true}$

(30)

Type: Boolean

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

$\label{eq31} \mbox{\rm true}$

(31)

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))

$\label{eq32}{{{\overline z}^{3}}+{{\left(-{3 \ z}+ 2 \right)}\ {{\overline z}^{2}}}+{{\left({3 \ {{z}^{2}}}+{4 \ z}\right)}\ {\overline z}}-{{z}^{3}}+{2 \ {{z}^{2}}}}\over 8$

(32)

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))

$\label{eq33} \mbox{\rm true}$

(33)

Type: Boolean

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

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

(34)

Type: Boolean

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

$\label{eq35}2 \ {\log \left({\sqrt{z \ {\overline z}}}\right)}$

(35)

Type: Expression(Integer)

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diff(t2,z)

$\label{eq36}1 \over z$

(36)

Type: Expression(Integer)

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diff(t2,conjugate z)

$\label{eq37}1 \over{\overline z}$

(37)

Type: Expression(Integer)