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SandBoxDiracDelta

last edited 9 years ago by Bill page

Edit detail for SandBoxDiracDelta revision 13 of 40

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Editor: Bill Page Time: 2014/08/11 20:31:38 GMT+0
Note:

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

changed:
-eval(realDirac,[x=0.1,y=1.0])::Float
-integrate(eval(realDirac,y=1),x=%minusInfinity..%plusInfinity,"noPole")
-limit(realDirac,y=0)
integrate(eval(realDirac,yy=1),x=%minusInfinity..%plusInfinity,"noPole")
integrate(eval(realDirac,yy=1/1000),x=%minusInfinity..%plusInfinity,"noPole")
limit(realDirac,yy=0)

changed:
-integrate(f(x)*diracDelta(x),x=%minusInfinity..%plusInfinity,"noPole")
--- y disappeared
---limit(%,y=0)
f(x)*diracDelta(x)
-- yy disappears?
-- limit(%,yy=0)
integrate(%,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

fricas

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

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

(1)

Type: Expression(Integer)

fricas

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

Type: Void

fricas

signum(x)

fricas

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

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

(2)

Type: Expression(Integer)

fricas

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

fricas

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)

fricas

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)

fricas

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

fricas

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

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

(5)

Type: Expression(Complex(Integer))

fricas

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)))

fricas

realDirac:=abs2sqrt(dirac2)/2

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

(7)

Type: Expression(Complex(Integer))

fricas

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

$\label{eq8}1$

(8)

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

fricas

integrate(eval(realDirac,yy=1/1000),x=%minusInfinity..%plusInfinity,"noPole")

$\label{eq9}1$

(9)

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

fricas

limit(realDirac,yy=0)

$\label{eq10}0$

(10)

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

fricas

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

   Compiled code for diracDelta has been cleared.
   1 old definition(s) deleted for function or rule diracDelta

Type: Void

fricas

f(x)==x+c

Type: Void

fricas

f(x)*diracDelta(x)

fricas

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

fricas

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

$\label{eq11}{{\left(x + c \right)}\ {{yy}^{2}}\ {\sqrt{{{yy}^{2}}+{{x}^{2}}}}}\over{{2 \ {{yy}^{4}}}+{4 \ {{x}^{2}}\ {{yy}^{2}}}+{2 \ {{x}^{4}}}}$

(11)

Type: Expression(Complex(Integer))

fricas

-- yy disappears?
-- limit(%,yy=0)
integrate(%,x=%minusInfinity..%plusInfinity,"noPole")

$\label{eq12}c$

(12)

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

Properties

fricas

signum(x^2)

fricas

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

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

(13)

Type: Expression(Integer)

fricas

diracDelta(x)^2

$\label{eq14}{{yy}^{4}}\over{{4 \ {{yy}^{6}}}+{{12}\ {{x}^{2}}\ {{yy}^{4}}}+{{12}\ {{x}^{4}}\ {{yy}^{2}}}+{4 \ {{x}^{6}}}}$

(14)

Type: Expression(Complex(Integer))

fricas

diracDelta(x^2)

fricas

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

$\label{eq15}{{{yy}^{2}}\ {\sqrt{{{yy}^{2}}+{{x}^{4}}}}}\over{{2 \ {{yy}^{4}}}+{4 \ {{x}^{4}}\ {{yy}^{2}}}+{2 \ {{x}^{8}}}}$

(15)

Type: Expression(Complex(Integer))

fricas

signum(x)*diracDelta(x)

$\label{eq16}{x \ {{yy}^{2}}\ {\sqrt{{{yy}^{2}}+{{x}^{2}}}}}\over{{\left({2 \ {{yy}^{4}}}+{4 \ {{x}^{2}}\ {{yy}^{2}}}+{2 \ {{x}^{4}}}\right)}\ {abs \left({x}\right)}}$

(16)

Type: Expression(Complex(Integer))

fricas

integrate(abs(x),x)

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

(17)

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

fricas

-- 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),...)

fricas

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

$\label{eq19}-{{{yy}^{2}}\over{{2 \ x \ {\sqrt{{{yy}^{2}}+{{x}^{2}}}}}-{2 \ {{yy}^{2}}}-{2 \ {{x}^{2}}}}}$

(19)

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

fricas

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

$\label{eq20}1$

(20)

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

fricas

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

$\label{eq21}0$

(21)

Type: Union(f1: 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(realDirac,[x=x1,y=1.0])::DFLOAT for x1 in X];

   Cannot convert from type Expression(Complex(DoubleFloat)) to 
      DoubleFloat for value
             +-----------+
           2 |  2
     0.5 yy \|yy  + 100.0
   -------------------------
     4           2
   yy  + 200.0 yy  + 10000.0

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"