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SandBoxDiracDelta

last edited 9 years ago by Bill page

Edit detail for SandBoxDiracDelta revision 10 of 40

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Editor: Bill page Time: 2014/08/11 13:39:50 GMT+0
Note: cleanup conjugate

changed:
-realDirac:=abs2sqrt dirac2
realDirac:=abs2sqrt(dirac2)/2

changed:
-integrate(realDirac,x=%minusInfinity..%plusInfinity,"noPole")
integrate((x+c)*realDirac,x=%minusInfinity..%plusInfinity,"noPole")

Load 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

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

$\label{eq1}\%x \over{abs \left({\%x}\right)}$

(1)

Type: Expression(Integer)

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signum(z)==eval(%signum,%x=z)

Type: Void

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signum(x)

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

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

(2)

Type: Expression(Integer)

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%diracDelta:=differentiate(signum(%x),%x)

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

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

(3)

Type: Expression(Integer)

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diracDelta(z)==eval(%diracDelta/2,%x=z)

Type: Void

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diracDelta(x)

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

$\label{eq4}{{{abs \left({x}\right)}^{2}}-{{x}^{2}}}\over{2 \ {{abs \left({x}\right)}^{3}}}$

(4)

Type: Expression(Integer)

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dirac2:=(diracDelta(x+%i*y)+diracDelta(x-%i*y))/2

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

$\label{eq5}{\left( \begin{array}{@{}l} \displaystyle {{\left({{abs \left({{i \ y}- x}\right)}^{2}}+{{y}^{2}}+{2 \ i \ x \ y}-{{x}^{2}}\right)}\ {{abs \left({{i \ y}+ x}\right)}^{3}}}+ \ \ \displaystyle {{{abs \left({{i \ y}- x}\right)}^{3}}\ {{abs \left({{i \ y}+ x}\right)}^{2}}}+ \ \ \displaystyle {{\left({{y}^{2}}-{2 \ i \ x \ y}-{{x}^{2}}\right)}\ {{abs \left({{i \ y}- x}\right)}^{3}}}$

(5)

Type: Expression(Complex(Integer))

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abs2sqrt:RewriteRule(Integer,Complex Integer,Expression(Complex Integer)):= rule abs(a+%i*b)==sqrt(a^2+b^2)

$\label{eq6}{abs \left({{i \ b}+ a}\right)}\mbox{\rm = =}{\sqrt{{{b}^{2}}+{{a}^{2}}}}$

(6)

Type: RewriteRule?(Integer,Complex(Integer),Expression(Complex(Integer)))

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realDirac:=abs2sqrt(dirac2)/2

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

(7)

Type: Expression(Complex(Integer))

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eval(realDirac,[x=0.1,y=1.0])::Float

$\label{eq8}0.4925926684_2078670082$

(8)

Type: Float

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integrate(eval(realDirac,y=1),x=%minusInfinity..%plusInfinity,"noPole")

$\label{eq9}1$

(9)

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

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limit(realDirac,y=0)

$\label{eq10}0$

(10)

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

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integrate((x+c)*realDirac,x=%minusInfinity..%plusInfinity,"noPole")

$\label{eq11}c$

(11)

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

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limit(%,y=0)

$\label{eq12}c$

(12)

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

Properties

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signum(x^2)

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

$\label{eq13}{{x}^{2}}\over{abs \left({{x}^{2}}\right)}$

(13)

Type: Expression(Integer)

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diracDelta(x)^2

$\label{eq14}{{{abs \left({x}\right)}^{4}}-{2 \ {{x}^{2}}\ {{abs \left({x}\right)}^{2}}}+{{x}^{4}}}\over{4 \ {{abs \left({x}\right)}^{6}}}$

(14)

Type: Expression(Integer)

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diracDelta(x^2)

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

$\label{eq15}{{{abs \left({{x}^{2}}\right)}^{2}}-{{x}^{4}}}\over{2 \ {{abs \left({{x}^{2}}\right)}^{3}}}$

(15)

Type: Expression(Integer)

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signum(x)*diracDelta(x)

$\label{eq16}{{x \ {{abs \left({x}\right)}^{2}}}-{{x}^{3}}}\over{2 \ {{abs \left({x}\right)}^{4}}}$

(16)

Type: Expression(Integer)

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integrate(abs(x),x)

$\label{eq17}\int^{ \displaystyle x}{{abs \left({\%A}\right)}\ {d \%A}}$

(17)

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

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-- expected x*abs(x)/2
integrate(signum(x),x)

$\label{eq18}\int^{ \displaystyle x}{{\%A \over{abs \left({\%A}\right)}}\ {d \%A}}$

(18)

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

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

$\label{eq19}\int^{ \displaystyle x}{{{{{abs \left({\%A}\right)}^{2}}-{{\%A}^{2}}}\over{2 \ {{abs \left({\%A}\right)}^{3}}}}\ {d \%A}}$

(19)

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

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-- expected signum(x)/2
--
integrate(diracDelta(x),x=minusInfinity..plusInfinity,"noPole")

$\label{eq20}\mbox{\tt "failed"}$

(20)

Type: Union(fail: failed,...)

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integrate(x*diracDelta(x),x=minusInfinity..plusInfinity,"noPole")

$\label{eq21}\mbox{\tt "failed"}$

(21)

Type: Union(fail: failed,...)

Bump function

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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(r1,[x=x1,y=1.0])::DFLOAT for x1 in X];

   Cannot convert from type Polynomial(DoubleFloat) to DoubleFloat for 
      value
   r1

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

   There are 2 exposed and 0 unexposed library operations named gnuDraw
      having 3 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                             )display op gnuDraw
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named 
      gnuDraw with argument type(s) 
                              List(DoubleFloat)
                                 Variable(Y)
                                   String

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

load "SandBoxDiracDelta1.dat"