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SandBoxDiracDelta

last edited 9 years ago by Bill page

Edit detail for SandBoxDiracDelta revision 1 of 40

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Editor: Bill page Time: 2014/08/09 19:15:04 GMT+0
Note:

changed:
-
Load new definition of derivative of abs(x)
\begin{axiom}
)lib FSPECX
\end{axiom}

\begin{axiom}
signum(x)==eval(differentiate(abs(%x),%x),%x=x)
signum(x)
diracDelta(x)==eval(differentiate(signum(%x),%x)/2,%x=x)
diracDelta(x)
realDirac:=(eval(diracDelta(z),z=x+%i*y)+eval(diracDelta(z),z=x-%i*y))/2
eval(realDirac,[x=0.1,y=1.0])
\end{axiom}

Properties
\begin{axiom}
signum(x^2)
diracDelta(x)^2
diracDelta(x^2)
signum(x)*diracDelta(x)
\end{axiom}

\begin{axiom}
integrate(abs(x),x)
-- expected x*abs(x)/2
integrate(signum(x),x)
-- expected abs(x)
integrate(diracDelta(x),x)
-- expected signum(x)/2
--
integrate(diracDelta(x),x=minusInfinity..plusInfinity,"noPole")
integrate(x*diracDelta(x),x=minusInfinity..plusInfinity,"noPole")
\end{axiom}

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

Type: Void

fricas

signum(x)

fricas

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

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

(1)

Type: Expression(Integer)

fricas

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

Type: Void

fricas

diracDelta(x)

fricas

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

fricas

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)

fricas

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

fricas

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))

fricas

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

$\label{eq4}0.9851853368_4157340165$

(4)

Type: Expression(Complex(Float))

Properties

fricas

signum(x^2)

fricas

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)

fricas

diracDelta(x^2)

fricas

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)

fricas

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)

fricas

integrate(abs(x),x)

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

(9)

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

fricas

-- 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),...)

fricas

-- 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),...)

fricas

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

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

(12)

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

fricas

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

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

(13)

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