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SandBoxWirtinger

Edit detail for SandBoxWirtinger revision 6 of 21

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Editor: Bill Page Time: 2014/08/15 22:53:02 GMT+0
Note: CR-calculus

changed:
-Ref.
Also called $\mathbb{CR}$-Calculus

Refs.

added:

- "The Complex Gradient Operator and the CR-Calculus":http://arxiv.org/abs/0906.4835
  by Ken Kreutz-Delgado

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

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

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

diff(sin(x),x)

fricas

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

$\label{eq2}\cos \left({x}\right)$

(2)

Type: Expression(Integer)

fricas

diff(sin(x),conjugate(x))

fricas

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

$\label{eq3}0$

(3)

Type: Expression(Integer)

fricas

diff(sqrt(x^2),x)

$\label{eq4}x \over{\sqrt{{x}^{2}}}$

(4)

Type: Expression(Integer)

fricas

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

$\label{eq5}0$

(5)

Type: Expression(Integer)

fricas

diff(z*conjugate(z),z)

fricas

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

$\label{eq6}\overline z$

(6)

Type: Expression(Integer)

fricas

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

$\label{eq7}z$

(7)

Type: Expression(Integer)

fricas

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

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

(8)

Type: Expression(Integer)

fricas

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

$\label{eq9}-{{\overline x}\over{4 \ x \ {\sqrt{x \ {\overline x}}}}}$

(9)

Type: Expression(Integer)

fricas

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

$\label{eq10}1 \over{4 \ {\sqrt{x \ {\overline x}}}}$

(10)

Type: Expression(Integer)

fricas

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

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

(11)

Type: Expression(Integer)

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)

$\label{eq12}1 \over 2$

(12)

Type: Expression(Integer)

fricas

diff(real(x),conjugate(x))

$\label{eq13}1 \over 2$

(13)

Type: Expression(Integer)

fricas

diff(imag(x),x)

fricas

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

$\label{eq14}1 \over 2$

(14)

Type: Expression(Integer)

fricas

diff(imag(x),conjugate(x))

$\label{eq15}-{1 \over 2}$

(15)

Type: Expression(Integer)

fricas

diff(abs(x)^2,x)

$\label{eq16}\overline x$

(16)

Type: Expression(Integer)