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SandBoxDiracDelta

last edited 9 years ago by Bill page

Edit detail for SandBoxDiracDelta revision 14 of 40

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Editor: Bill Page Time: 2014/08/11 22:01:08 GMT+0
Note:

changed:
-Load new definition of derivative of abs(x)
New definition of derivative of abs(x)

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

added:
Derivatives

changed:
-%signum:=differentiate(abs(%x),%x)
-signum(z)==eval(%signum,%x=z)
%signum:=differentiate(abs(%z),%z)
%diracDelta:=differentiate(%signum,%z)/2
\end{axiom}

Distributions
\begin{axiom}
realSignum:=abs2sqrt(eval(%signum,%z=a+%i*b)+eval(%signum,%z=a-%i*b))/2
signum(x)==eval(realSignum,a=x)

changed:
-%diracDelta:=differentiate(signum(%x),%x)
-diracDelta(z)==eval(%diracDelta/2,%x=z)
--
realDirac:=abs2sqrt(eval(%diracDelta,%z=a+%i*b)+eval(%diracDelta, %z=a-%i*b))/4
diracDelta(x)==eval(realDirac,a=x)

added:
Properties

changed:
-dirac2:=(diracDelta(x+%i*yy)+diracDelta(x-%i*yy))/2
-abs2sqrt:RewriteRule(Integer,Complex Integer,Expression(Complex Integer)):= rule abs(a+%i*b)==sqrt(a^2+b^2)
-realDirac:=abs2sqrt(dirac2)/2
-integrate(eval(realDirac,yy=1),x=%minusInfinity..%plusInfinity,"noPole")
-integrate(eval(realDirac,yy=1/1000),x=%minusInfinity..%plusInfinity,"noPole")
-limit(realDirac,yy=0)
-diracDelta(z)==eval(realDirac,x=z)
signum(x)^2
limit(%,b=0)
diracDelta(x)^2
limit(%,b=0)
-- b disappears?
integrate(diracDelta(x),x=%minusInfinity..%plusInfinity,"noPole")
limit(%,b=0)
-- generally

removed:
--- yy disappears?
--- limit(%,yy=0)

added:
signum(x)*diracDelta(x)
limit(%,b=0)
diracDelta(x^2)
limit(%,b=0)

changed:
-Properties
-\begin{axiom}
-signum(x^2)
-diracDelta(x)^2
-diracDelta(x^2)
-signum(x)*diracDelta(x)
-\end{axiom}
-
Problems?

added:
eval(abs2sqrt(abs(a+%i*b)/2+abs(a-%i*b)/2),a=x)
integrate(%,x)
limit(%,b=0)
integrate(%signum,%z)
-- expected abs(x)

changed:
--- expected abs(x)
limit(%,b=0)
integrate(%diracDelta,%z)
-- expected
signum(x)/2

changed:
--- expected signum(x)/2
---
-integrate(diracDelta(x),x=%minusInfinity..%plusInfinity,"noPole")
-integrate(x*diracDelta(x),x=%minusInfinity..%plusInfinity,"noPole")
limit(%,b=0)

changed:
-Y:=[eval(realDirac,[x=x1,y=1.0])::DFLOAT for x1 in X];
Y:=[eval(diracDelta(x),b=1.0)::DFLOAT for x in X];

New definition of derivative of abs(x)

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)))

Derivatives

fricas

%signum:=differentiate(abs(%z),%z)

$\label{eq2}\%z \over{abs \left({\%z}\right)}$

(2)

Type: Expression(Integer)

fricas

%diracDelta:=differentiate(%signum,%z)/2

$\label{eq3}{{{abs \left({\%z}\right)}^{2}}-{{\%z}^{2}}}\over{2 \ {{abs \left({\%z}\right)}^{3}}}$

(3)

Type: Expression(Integer)

Distributions

fricas

realSignum:=abs2sqrt(eval(%signum,%z=a+%i*b)+eval(%signum,%z=a-%i*b))/2

$\label{eq4}{a \ {\sqrt{{{b}^{2}}+{{a}^{2}}}}}\over{{{b}^{2}}+{{a}^{2}}}$

(4)

Type: Expression(Complex(Integer))

fricas

signum(x)==eval(realSignum,a=x)

Type: Void

fricas

signum(x)

fricas

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

$\label{eq5}{x \ {\sqrt{{{x}^{2}}+{{b}^{2}}}}}\over{{{x}^{2}}+{{b}^{2}}}$

(5)

Type: Expression(Complex(Integer))

fricas

--
realDirac:=abs2sqrt(eval(%diracDelta,%z=a+%i*b)+eval(%diracDelta, %z=a-%i*b))/4

$\label{eq6}{{{b}^{2}}\ {\sqrt{{{b}^{2}}+{{a}^{2}}}}}\over{{2 \ {{b}^{4}}}+{4 \ {{a}^{2}}\ {{b}^{2}}}+{2 \ {{a}^{4}}}}$

(6)

Type: Expression(Complex(Integer))

fricas

diracDelta(x)==eval(realDirac,a=x)

Type: Void

fricas

diracDelta(x)

fricas

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

$\label{eq7}{{{b}^{2}}\ {\sqrt{{{x}^{2}}+{{b}^{2}}}}}\over{{2 \ {{x}^{4}}}+{4 \ {{b}^{2}}\ {{x}^{2}}}+{2 \ {{b}^{4}}}}$

(7)

Type: Expression(Complex(Integer))

Properties

fricas

signum(x)^2

$\label{eq8}{{x}^{2}}\over{{{x}^{2}}+{{b}^{2}}}$

(8)

Type: Expression(Complex(Integer))

fricas

limit(%,b=0)

$\label{eq9}1$

(9)

Type: Union(OrderedCompletion?(Expression(Complex(Integer))),...)

fricas

diracDelta(x)^2

$\label{eq10}{{b}^{4}}\over{{4 \ {{x}^{6}}}+{{12}\ {{b}^{2}}\ {{x}^{4}}}+{{1 2}\ {{b}^{4}}\ {{x}^{2}}}+{4 \ {{b}^{6}}}}$

(10)

Type: Expression(Complex(Integer))

fricas

limit(%,b=0)

$\label{eq11}0$

(11)

Type: Union(OrderedCompletion?(Expression(Complex(Integer))),...)

fricas

-- b disappears?
integrate(diracDelta(x),x=%minusInfinity..%plusInfinity,"noPole")

$\label{eq12}1$

(12)

Type: Union(f1: OrderedCompletion?(Expression(Complex(Integer))),...)

fricas

limit(%,b=0)

$\label{eq13}1$

(13)

Type: Union(OrderedCompletion?(Expression(Integer)),...)

fricas

-- generally
f(x)==x+c

Type: Void

fricas

f(x)*diracDelta(x)

fricas

Compiling function f with type Variable(x) -> Polynomial(Integer)

$\label{eq14}{{\left({{{b}^{2}}\ x}+{{{b}^{2}}\ c}\right)}\ {\sqrt{{{x}^{2}}+{{b}^{2}}}}}\over{{2 \ {{x}^{4}}}+{4 \ {{b}^{2}}\ {{x}^{2}}}+{2 \ {{b}^{4}}}}$

(14)

Type: Expression(Complex(Integer))

fricas

integrate(%,x=%minusInfinity..%plusInfinity,"noPole")

$\label{eq15}c$

(15)

Type: Union(f1: OrderedCompletion?(Expression(Complex(Integer))),...)

fricas

signum(x)*diracDelta(x)

$\label{eq16}{{{b}^{2}}\ x}\over{{2 \ {{x}^{4}}}+{4 \ {{b}^{2}}\ {{x}^{2}}}+{2 \ {{b}^{4}}}}$

(16)

Type: Expression(Complex(Integer))

fricas

limit(%,b=0)

$\label{eq17}0$

(17)

Type: Union(OrderedCompletion?(Expression(Complex(Integer))),...)

fricas

diracDelta(x^2)

fricas

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

$\label{eq18}{{{b}^{2}}\ {\sqrt{{{x}^{4}}+{{b}^{2}}}}}\over{{2 \ {{x}^{8}}}+{4 \ {{b}^{2}}\ {{x}^{4}}}+{2 \ {{b}^{4}}}}$

(18)

Type: Expression(Complex(Integer))

fricas

limit(%,b=0)

$\label{eq19}0$

(19)

Type: Union(OrderedCompletion?(Expression(Complex(Integer))),...)

Problems?

fricas

integrate(abs(x),x)

$\label{eq20}\int^{ \displaystyle x}{{abs \left({\%C}\right)}\ {d \%C}}$

(20)

Type: Union(Expression(Integer),...)

fricas

-- expected x*abs(x)/2
eval(abs2sqrt(abs(a+%i*b)/2+abs(a-%i*b)/2),a=x)

$\label{eq21}\sqrt{{{x}^{2}}+{{b}^{2}}}$

(21)

Type: Expression(Complex(Integer))

fricas

integrate(%,x)

$\label{eq22}{\left( \begin{array}{@{}l} \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle -{2 \ {{b}^{2}}\ x \ {\sqrt{{{x}^{2}}+{{b}^{2}}}}}+ \ \ \displaystyle {2 \ {{b}^{2}}\ {{x}^{2}}}+{{b}^{4}}$

(22)

Type: Union(Expression(Complex(Integer)),...)

fricas

limit(%,b=0)

$\label{eq23}-{{{x}^{2}}\over 2}$

(23)

Type: Union(OrderedCompletion?(Expression(Complex(Integer))),...)

fricas

integrate(%signum,%z)

$\label{eq24}\int^{ \displaystyle \%z}{{\%C \over{abs \left({\%C}\right)}}\ {d \%C}}$

(24)

Type: Union(Expression(Integer),...)

fricas

-- expected abs(x)
integrate(signum(x),x)

$\label{eq25}{-{x \ {\sqrt{{{x}^{2}}+{{b}^{2}}}}}+{{x}^{2}}+{{b}^{2}}}\over{{\sqrt{{{x}^{2}}+{{b}^{2}}}}- x}$

(25)

Type: Union(Expression(Complex(Integer)),...)

fricas

limit(%,b=0)

$\label{eq26}- x$

(26)

Type: Union(OrderedCompletion?(Expression(Complex(Integer))),...)

fricas

integrate(%diracDelta,%z)

$\label{eq27}\int^{ \displaystyle \%z}{{{{{abs \left({\%C}\right)}^{2}}-{{\%C}^{2}}}\over{2 \ {{abs \left({\%C}\right)}^{3}}}}\ {d \%C}}$

(27)

Type: Union(Expression(Integer),...)

fricas

-- expected
signum(x)/2

$\label{eq28}{x \ {\sqrt{{{x}^{2}}+{{b}^{2}}}}}\over{{2 \ {{x}^{2}}}+{2 \ {{b}^{2}}}}$

(28)

Type: Expression(Complex(Integer))

fricas

integrate(diracDelta(x),x)

$\label{eq29}-{{{b}^{2}}\over{{2 \ x \ {\sqrt{{{x}^{2}}+{{b}^{2}}}}}-{2 \ {{x}^{2}}}-{2 \ {{b}^{2}}}}}$

(29)

Type: Union(Expression(Complex(Integer)),...)

fricas

limit(%,b=0)

$\label{eq30}0$

(30)

Type: Union(OrderedCompletion?(Expression(Complex(Integer))),...)

Representing diracDelta as a series of bump functions

fricas

)lib GDRAW

   GnuDraw is now explicitly exposed in frame initial 
   GnuDraw will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/GDRAW.NRLIB/GDRAW
X:=[(x/10)::DFLOAT for x in -100..100 by 2];

Type: List(DoubleFloat?)

fricas

Y:=[eval(diracDelta(x),b=1.0)::DFLOAT for x in X];

fricas

Compiling function diracDelta with type DoubleFloat -> Expression(
      Complex(DoubleFloat))

Type: List(DoubleFloat?)

fricas

gnuDraw(X,Y,"SandBoxDiracDelta1.dat")

   Graph data being transmitted to the viewport manager...
   FriCAS2D data being transmitted to the viewport manager...

Type: Void

load "SandBoxDiracDelta1.dat"