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Edit detail for SandBoxWirtinger revision 7 of 21

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Editor: Bill Page Time: 2014/08/15 23:37:17 GMT+0
Note: CR-calculus

added:

- "Appendix A: Wirtinger Calculus":http://onlinelibrary.wiley.com/doi/10.1002/0471439002.app1/summary
  by Robert F. H. Fischer

removed:
-

added:
test(diff(z,z)=1)
test(diff(conjugate z,conjugate z)=1)
test(diff(z,conjugate z)=0)
test(diff(conjugate z,z)=0)

changed:
-diff(sin(x),conjugate(x))
diff(sin(x),conjugate x)

changed:
-diff(sqrt(x^2),conjugate(x))
diff(sqrt(x^2),conjugate x)

changed:
-diff(z*conjugate(z),conjugate(z))
diff(z*conjugate(z),conjugate z)

changed:
-diff(diff(sqrt(x*conjugate(x)),x),conjugate(x))
-diff(sqrt(x*conjugate(x)),conjugate(x))
diff(diff(sqrt(x*conjugate(x)),x),conjugate x)
diff(sqrt(x*conjugate(x)),conjugate x)

added:
Non-holomorphic functions

changed:
-diff(real(x),conjugate(x))
diff(real(x),conjugate x)

changed:
-diff(imag(x),conjugate(x))
diff(imag(x),conjugate x)

added:

Functions are assumed holomorphic
\begin{axiom}
f:=operator 'f
diff(f(x),conjugate x)
diff(f(x,conjugate x),x)
diff(f(x,conjugate x),conjugate x)
test(diff(conjugate f(z,conjugate z),conjugate z) = conjugate diff(f(z,conjugate z),z))
test(diff(conjugate f(z,conjugate z),z) = conjugate diff(f(z,conjugate z),conjugate z))
\end{axiom}

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

fricas

)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)))

fricas

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

Type: Void

fricas

test(diff(z,z)=1)

fricas

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

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

(2)

Type: Boolean

fricas

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

fricas

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

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

(3)

Type: Boolean

fricas

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

fricas

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

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

(4)

Type: Boolean

fricas

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

fricas

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

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

(5)

Type: Boolean

fricas

diff(sin(x),x)

fricas

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

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

(6)

Type: Expression(Integer)

fricas

diff(sin(x),conjugate x)

$\label{eq7}0$

(7)

Type: Expression(Integer)

fricas

diff(sqrt(x^2),x)

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

(8)

Type: Expression(Integer)

fricas

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

$\label{eq9}0$

(9)

Type: Expression(Integer)

fricas

diff(z*conjugate(z),z)

$\label{eq10}\overline z$

(10)

Type: Expression(Integer)

fricas

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

$\label{eq11}z$

(11)

Type: Expression(Integer)

fricas

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

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

(12)

Type: Expression(Integer)

fricas

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

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

(13)

Type: Expression(Integer)

fricas

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

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

(14)

Type: Expression(Integer)

fricas

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

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

(15)

Type: Expression(Integer)

Non-holomorphic functions

fricas

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

Type: Void

fricas

imag(x) == (x-conjugate(x))/2

Type: Void

fricas

diff(real(x),x)

fricas

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

$\label{eq16}1 \over 2$

(16)

Type: Expression(Integer)

fricas

diff(real(x),conjugate x)

$\label{eq17}1 \over 2$

(17)

Type: Expression(Integer)

fricas

diff(imag(x),x)

fricas

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

$\label{eq18}1 \over 2$

(18)

Type: Expression(Integer)

fricas

diff(imag(x),conjugate x)

$\label{eq19}-{1 \over 2}$

(19)

Type: Expression(Integer)

fricas

diff(abs(x)^2,x)

$\label{eq20}\overline x$

(20)

Type: Expression(Integer)

Functions are assumed holomorphic

fricas

f:=operator 'f

$\label{eq21}f$

(21)

Type: BasicOperator?

fricas

diff(f(x),conjugate x)

$\label{eq22}0$

(22)

Type: Expression(Integer)

fricas

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

$\label{eq23}{f_{, 1}}\left({x , \:{\overline x}}\right)$

(23)

Type: Expression(Integer)

fricas

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

$\label{eq24}{f_{, 2}}\left({x , \:{\overline x}}\right)$

(24)

Type: Expression(Integer)

fricas

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

$\label{eq25} \mbox{\rm true}$

(25)

Type: Boolean

fricas

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

$\label{eq26} \mbox{\rm true}$

(26)

Type: Boolean