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Wirtinger Derivatives or Wirtinger Calculus (also called -Calculus)

Reference

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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,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)
 (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)
 (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)
 (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)
 (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)
 (5)
Type: Expression(Integer)
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diff(sin(x),conjugate x)
 (6)
Type: Expression(Integer)
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diff(sqrt(x^2),x)
 (7)
Type: Expression(Integer)
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diff(sqrt(x^2),conjugate x)
 (8)
Type: Expression(Integer)
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diff(z*conjugate(z),z)
 (9)
Type: Expression(Integer)
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diff(z*conjugate(z),conjugate z)
 (10)
Type: Expression(Integer)
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diff(sqrt(x*conjugate(x)),x)
 (11)
Type: Expression(Integer)
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diff(%,x)
 (12)
Type: Expression(Integer)
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diff(%,x)
 (13)
Type: Expression(Integer)
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diff(diff(diff(abs(x),x),x),x)
 (14)
Type: Expression(Integer)
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_rule(abs(x),sqrt(x*conjugate(x))) %
 (15)
Type: Expression(Integer)
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diff(sqrt(x*conjugate(x)),conjugate x)
 (16)
Type: Expression(Integer)
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conjugate(diff(conjugate(sqrt(x*conjugate(x))),x))
 (17)
Type: Expression(Integer)
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diff(%,conjugate x)
 (18)
Type: Expression(Integer)
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diff(%,conjugate x)
 (19)
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)
 (20)
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))
 (21)
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))
 (22)
Type: Expression(Complex(Integer))
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eval(%,[conjugate(a)=a,conjugate(b)=b])
 (23)
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))
 (24)
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))
 (25)
Type: Expression(Complex(Integer))
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eval(%,[conjugate(a)=a,conjugate(b)=b])
 (26)
Type: Expression(Complex(Integer))
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diff(real(x),x)
 (27)
Type: Expression(Integer)
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diff(real(x),conjugate x)
 (28)
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))
 (29)
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))
 (30)
Type: Expression(Complex(Integer))
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diff(abs(x)^2,x)
 (31)
Type: Expression(Integer)

Functions are assumed holomorphic

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f:=operator 'f
 (32)
Type: BasicOperator?
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diff(c*f(x),x)
 (33)
Type: Expression(Integer)
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diff(c*f(x),conjugate x)
 (34)
Type: Expression(Integer)
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diff(f(x,conjugate x),x)
 (35)
Type: Expression(Integer)
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diff(f(x,conjugate x),conjugate x)
 (36)
Type: Expression(Integer)
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test(diff(conjugate f(z,conjugate z),conjugate z) = conjugate diff(f(z,conjugate z),z))
 (37)
Type: Boolean
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test(diff(conjugate f(z,conjugate z),z) = conjugate diff(f(z,conjugate z),conjugate z))
 (38)
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))
 (39)
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))
 (40)
Type: Boolean
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(diff(t1,conjugate z) = 1/2*z+1/2*conjugate(z)-3/8*(z-conjugate(z))^2)
 (41)
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)
 (42)
Type: Boolean
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t2:=2*log(sqrt(z*conjugate(z)))
 (43)
Type: Expression(Integer)
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diff(t2,z)
 (44)
Type: Expression(Integer)
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diff(t2,conjugate z)
 (45)
Type: Expression(Integer)
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diff(sqrt(x*conjugate(x))/x,x)+diff(sqrt(x*conjugate(x))/x,conjugate x)
 (46)
Type: Expression(Integer)
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eval(%,conjugate(x)=x)
 (47)
Type: Expression(Integer)

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