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SandBoxDiracDelta

last edited 10 years ago by Bill page

Edit detail for SandBoxDiracDelta revision 4 of 40

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Editor: Bill page Time: 2014/08/10 04:33:58 GMT+0
Note:

changed:
-conj:RewriteRule(Integer,Complex Integer,Expression(Complex Integer)):= rule %i==-%i
--conj1:RewriteRule(Integer,Complex Integer,Expression(Complex Integer)):= rule %i==-sqrt(-1)
--conj1(a+%i*b)

changed:
-X:=[(x/100)::DFLOAT for x in 1..50 by 5]
-Y:=[eval(realDirac,[x=x1,y=1.0])::DFLOAT for x1 in X]
X:=[(x/10)::DFLOAT for x in -100..100 by 2];
Y:=[eval(realDirac,[x=x1,y=1.0])::DFLOAT for x1 in X];

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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abs2sqrt:RewriteRule(Integer,Complex Integer,Expression(Complex Integer)):= rule abs(x)==sqrt(x^2)

$\label{eq4}{abs \left({x}\right)}\mbox{\rm = =}{\sqrt{{x}^{2}}}$

(4)

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

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--conj1:RewriteRule(Integer,Complex Integer,Expression(Complex Integer)):= rule %i==-sqrt(-1)
--conj1(a+%i*b)
abs2sqrt(abs(a+%i*b))

$\label{eq5}\sqrt{-{{b}^{2}}+{2 \ i \ a \ b}+{{a}^{2}}}$

(5)

Type: Expression(Complex(Integer))

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abs2sqrt realDirac

$\label{eq6}0$

(6)

Type: Expression(Complex(Integer))

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

$\label{eq7}0.9851853368_4157340165$

(7)

Type: Float

Properties

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signum(x^2)

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

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

(8)

Type: Expression(Integer)

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diracDelta(x)^2

$\label{eq9}{{{abs \left({x}\right)}^{4}}-{2 \ {{x}^{2}}\ {{abs \left({x}\right)}^{2}}}+{{x}^{4}}}\over{4 \ {{abs \left({x}\right)}^{6}}}$

(9)

Type: Expression(Integer)

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diracDelta(x^2)

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

$\label{eq10}{{{abs \left({{x}^{2}}\right)}^{2}}-{{x}^{4}}}\over{2 \ {{abs \left({{x}^{2}}\right)}^{3}}}$

(10)

Type: Expression(Integer)

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signum(x)*diracDelta(x)

$\label{eq11}{{x \ {{abs \left({x}\right)}^{2}}}-{{x}^{3}}}\over{2 \ {{abs \left({x}\right)}^{4}}}$

(11)

Type: Expression(Integer)

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integrate(abs(x),x)

$\label{eq12}\int^{ \displaystyle x}{{abs \left({\%A}\right)}\ {d \%A}}$

(12)

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

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

$\label{eq13}\int^{ \displaystyle x}{{\%A \over{abs \left({\%A}\right)}}\ {d \%A}}$

(13)

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

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

$\label{eq14}\int^{ \displaystyle x}{{{{{abs \left({\%A}\right)}^{2}}-{{\%A}^{2}}}\over{2 \ {{abs \left({\%A}\right)}^{3}}}}\ {d \%A}}$

(14)

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

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

$\label{eq15}\mbox{\tt "failed"}$

(15)

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

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

$\label{eq16}\mbox{\tt "failed"}$

(16)

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
X:=[(x/10)::DFLOAT for x in -100..100 by 2];

Type: List(DoubleFloat?)

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

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"