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SandBoxDiracDelta

last edited 10 years ago by Bill page

Edit detail for SandBoxDiracDelta revision 2 of 40

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Editor: Bill page Time: 2014/08/09 19:35:40 GMT+0
Note:

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

added:

\begin{axiom}
)lib GDRAW
t1(x:Float):Float == real eval(realDirac,y=1.0)
t1(10.0)
gnuDraw(t1, -10..10,"SandBoxDiracDelta1.dat")
\end{axiom}

\begin{gnuplot}
load "SandBoxDiracDelta1.dat"
\end{gnuplot}

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(x)==eval(differentiate(abs(%x),%x),%x=x)

Type: Void

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signum(x)

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

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

(1)

Type: Expression(Integer)

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diracDelta(x)==eval(differentiate(signum(%x),%x)/2,%x=x)

Type: Void

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diracDelta(x)

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

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

$\label{eq2}{{{abs \left({x}\right)}^{2}}-{{x}^{2}}}\over{2 \ {{abs \left({x}\right)}^{3}}}$

(2)

Type: Expression(Integer)

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realDirac:=(eval(diracDelta(z),z=x+%i*y)+eval(diracDelta(z),z=x-%i*y))/2

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

$\label{eq3}{\left( \begin{array}{@{}l} \displaystyle {{\left({{abs \left({{i \ y}- x}\right)}^{2}}+{{y}^{2}}+{2 \ i \ x \ y}-{{x}^{2}}\right)}\ {{abs \left({{i \ y}+ x}\right)}^{3}}}+ \ \ \displaystyle {{{abs \left({{i \ y}- x}\right)}^{3}}\ {{abs \left({{i \ y}+ x}\right)}^{2}}}+ \ \ \displaystyle {{\left({{y}^{2}}-{2 \ i \ x \ y}-{{x}^{2}}\right)}\ {{abs \left({{i \ y}- x}\right)}^{3}}}$

(3)

Type: Expression(Complex(Integer))

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

$\label{eq4}0.9851853368_4157340165$

(4)

Type: Float

Properties

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signum(x^2)

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

$\label{eq5}{{x}^{2}}\over{abs \left({{x}^{2}}\right)}$

(5)

Type: Expression(Integer)

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diracDelta(x)^2

$\label{eq6}{{{abs \left({x}\right)}^{4}}-{2 \ {{x}^{2}}\ {{abs \left({x}\right)}^{2}}}+{{x}^{4}}}\over{4 \ {{abs \left({x}\right)}^{6}}}$

(6)

Type: Expression(Integer)

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diracDelta(x^2)

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

$\label{eq7}{{{abs \left({{x}^{2}}\right)}^{2}}-{{x}^{4}}}\over{2 \ {{abs \left({{x}^{2}}\right)}^{3}}}$

(7)

Type: Expression(Integer)

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signum(x)*diracDelta(x)

$\label{eq8}{{x \ {{abs \left({x}\right)}^{2}}}-{{x}^{3}}}\over{2 \ {{abs \left({x}\right)}^{4}}}$

(8)

Type: Expression(Integer)

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integrate(abs(x),x)

$\label{eq9}\int^{ \displaystyle x}{{abs \left({\%A}\right)}\ {d \%A}}$

(9)

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

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

$\label{eq10}\int^{ \displaystyle x}{{\%A \over{abs \left({\%A}\right)}}\ {d \%A}}$

(10)

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

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

$\label{eq11}\int^{ \displaystyle x}{{{{{abs \left({\%A}\right)}^{2}}-{{\%A}^{2}}}\over{2 \ {{abs \left({\%A}\right)}^{3}}}}\ {d \%A}}$

(11)

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

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

$\label{eq12}\mbox{\tt "failed"}$

(12)

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

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

$\label{eq13}\mbox{\tt "failed"}$

(13)

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

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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
t1(x:Float):Float == real eval(realDirac,y=1.0)

   Function declaration t1 : Float -> Float has been added to 
      workspace.

Type: Void

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t1(10.0)

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Compiling function t1 with type Float -> Float 
   Conversion failed in the compiled user function t1 .

   Cannot convert from type Expression(Float) to Float for value
       +--------+
       | 2
      \|x  + 1.0
   -----------------
    4        2
   x  + 2.0 x  + 1.0

gnuDraw(t1, -10..10,"SandBoxDiracDelta1.dat")

   There are 1 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) 
                             FunctionCalled(t1)
                              Segment(Integer)
                                   String

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