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SandBoxDiracDelta

last edited 10 years ago by Bill page

Edit detail for SandBoxDiracDelta revision 15 of 40

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Editor: Bill Page Time: 2014/08/11 22:43:29 GMT+0
Note:

added:
diracDelta(1)
limit(%,b=0)
diracDelta(-1)
limit(%,b=0)
diracDelta(0)
limit(%,b=0)
signum(1)
limit(%,b=0)
signum(-1)
limit(%,b=0)
signum(0)
limit(%,b=0)
--

added:
--

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

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

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

(2)

Type: Expression(Integer)

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%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

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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))

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

Type: Void

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signum(x)

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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))

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--
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))

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

Type: Void

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diracDelta(x)

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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

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diracDelta(1)

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

$\label{eq8}{{{b}^{2}}\ {\sqrt{{{b}^{2}}+ 1}}}\over{{2 \ {{b}^{4}}}+{4 \ {{b}^{2}}}+ 2}$

(8)

Type: Expression(Complex(Integer))

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limit(%,b=0)

$\label{eq9}0$

(9)

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

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diracDelta(-1)

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

$\label{eq10}{{{b}^{2}}\ {\sqrt{{{b}^{2}}+ 1}}}\over{{2 \ {{b}^{4}}}+{4 \ {{b}^{2}}}+ 2}$

(10)

Type: Expression(Complex(Integer))

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limit(%,b=0)

$\label{eq11}0$

(11)

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

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diracDelta(0)

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

$\label{eq12}{\sqrt{{b}^{2}}}\over{2 \ {{b}^{2}}}$

(12)

Type: Expression(Complex(Integer))

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limit(%,b=0)

$\label{eq13}+ \infty$

(13)

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

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signum(1)

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

$\label{eq14}{\sqrt{{{b}^{2}}+ 1}}\over{{{b}^{2}}+ 1}$

(14)

Type: Expression(Complex(Integer))

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limit(%,b=0)

$\label{eq15}1$

(15)

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

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signum(-1)

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

$\label{eq16}-{{\sqrt{{{b}^{2}}+ 1}}\over{{{b}^{2}}+ 1}}$

(16)

Type: Expression(Complex(Integer))

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limit(%,b=0)

$\label{eq17}- 1$

(17)

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

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signum(0)

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

$\label{eq18}0$

(18)

Type: Expression(Complex(Integer))

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limit(%,b=0)

$\label{eq19}0$

(19)

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

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--
signum(x)^2

$\label{eq20}{{x}^{2}}\over{{{x}^{2}}+{{b}^{2}}}$

(20)

Type: Expression(Complex(Integer))

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limit(%,b=0)

$\label{eq21}1$

(21)

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

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diracDelta(x)^2

$\label{eq22}{{b}^{4}}\over{{4 \ {{x}^{6}}}+{{12}\ {{b}^{2}}\ {{x}^{4}}}+{{1 2}\ {{b}^{4}}\ {{x}^{2}}}+{4 \ {{b}^{6}}}}$

(22)

Type: Expression(Complex(Integer))

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limit(%,b=0)

$\label{eq23}0$

(23)

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

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

$\label{eq24}1$

(24)

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

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limit(%,b=0)

$\label{eq25}1$

(25)

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

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

Type: Void

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f(x)*diracDelta(x)

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

$\label{eq26}{{\left({{{b}^{2}}\ x}+{{{b}^{2}}\ c}\right)}\ {\sqrt{{{x}^{2}}+{{b}^{2}}}}}\over{{2 \ {{x}^{4}}}+{4 \ {{b}^{2}}\ {{x}^{2}}}+{2 \ {{b}^{4}}}}$

(26)

Type: Expression(Complex(Integer))

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

$\label{eq27}c$

(27)

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

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--
signum(x)*diracDelta(x)

$\label{eq28}{{{b}^{2}}\ x}\over{{2 \ {{x}^{4}}}+{4 \ {{b}^{2}}\ {{x}^{2}}}+{2 \ {{b}^{4}}}}$

(28)

Type: Expression(Complex(Integer))

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limit(%,b=0)

$\label{eq29}0$

(29)

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

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diracDelta(x^2)

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

$\label{eq30}{{{b}^{2}}\ {\sqrt{{{x}^{4}}+{{b}^{2}}}}}\over{{2 \ {{x}^{8}}}+{4 \ {{b}^{2}}\ {{x}^{4}}}+{2 \ {{b}^{4}}}}$

(30)

Type: Expression(Complex(Integer))

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limit(%,b=0)

$\label{eq31}0$

(31)

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

Problems?

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integrate(abs(x),x)

$\label{eq32}\int^{ \displaystyle x}{{abs \left({\%C}\right)}\ {d \%C}}$

(32)

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

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

$\label{eq33}\sqrt{{{x}^{2}}+{{b}^{2}}}$

(33)

Type: Expression(Complex(Integer))

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integrate(%,x)

$\label{eq34}{\left( \begin{array}{@{}l} \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle -{2 \ {{b}^{2}}\ x \ {\sqrt{{{x}^{2}}+{{b}^{2}}}}}+ \ \ \displaystyle {2 \ {{b}^{2}}\ {{x}^{2}}}+{{b}^{4}}$

(34)

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

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limit(%,b=0)

$\label{eq35}-{{{x}^{2}}\over 2}$

(35)

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

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integrate(%signum,%z)

$\label{eq36}\int^{ \displaystyle \%z}{{\%C \over{abs \left({\%C}\right)}}\ {d \%C}}$

(36)

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

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

$\label{eq37}{-{x \ {\sqrt{{{x}^{2}}+{{b}^{2}}}}}+{{x}^{2}}+{{b}^{2}}}\over{{\sqrt{{{x}^{2}}+{{b}^{2}}}}- x}$

(37)

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

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limit(%,b=0)

$\label{eq38}- x$

(38)

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

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integrate(%diracDelta,%z)

$\label{eq39}\int^{ \displaystyle \%z}{{{{{abs \left({\%C}\right)}^{2}}-{{\%C}^{2}}}\over{2 \ {{abs \left({\%C}\right)}^{3}}}}\ {d \%C}}$

(39)

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

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-- expected
signum(x)/2

$\label{eq40}{x \ {\sqrt{{{x}^{2}}+{{b}^{2}}}}}\over{{2 \ {{x}^{2}}}+{2 \ {{b}^{2}}}}$

(40)

Type: Expression(Complex(Integer))

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integrate(diracDelta(x),x)

$\label{eq41}-{{{b}^{2}}\over{{2 \ x \ {\sqrt{{{x}^{2}}+{{b}^{2}}}}}-{2 \ {{x}^{2}}}-{2 \ {{b}^{2}}}}}$

(41)

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

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limit(%,b=0)

$\label{eq42}0$

(42)

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

Representing diracDelta as a series of bump functions

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)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?)

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

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

Type: List(DoubleFloat?)

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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"