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SandBoxDiracDelta

last edited 10 years ago by Bill page

Edit detail for SandBoxDiracDelta revision 3 of 40

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Editor: Bill page Time: 2014/08/10 03:48:59 GMT+0
Note:

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

changed:
-t1(x:Float):Float == real eval(realDirac,y=1.0)
-t1(10.0)
-gnuDraw(t1, -10..10,"SandBoxDiracDelta1.dat")
X:=[(x/100)::DFLOAT for x in 1..50 by 5]
Y:=[eval(realDirac,[x=x1,y=1.0])::DFLOAT for x1 in X]
gnuDraw(X,Y,"SandBoxDiracDelta1.dat")

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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conj:RewriteRule(Integer,Complex Integer,Expression(Complex Integer)):= rule %i==-%i

$\label{eq5}i \mbox{\rm = =}- i$

(5)

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

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abs2sqrt(abs(a+%i*b))

$\label{eq6}\sqrt{-{{b}^{2}}+{2 \ i \ a \ b}+{{a}^{2}}}$

(6)

Type: Expression(Complex(Integer))

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abs2sqrt realDirac

$\label{eq7}0$

(7)

Type: Expression(Complex(Integer))

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

$\label{eq8}0.9851853368_4157340165$

(8)

Type: Float

Properties

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signum(x^2)

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

$\label{eq9}{{x}^{2}}\over{abs \left({{x}^{2}}\right)}$

(9)

Type: Expression(Integer)

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diracDelta(x)^2

$\label{eq10}{{{abs \left({x}\right)}^{4}}-{2 \ {{x}^{2}}\ {{abs \left({x}\right)}^{2}}}+{{x}^{4}}}\over{4 \ {{abs \left({x}\right)}^{6}}}$

(10)

Type: Expression(Integer)

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diracDelta(x^2)

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

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

(11)

Type: Expression(Integer)

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signum(x)*diracDelta(x)

$\label{eq12}{{x \ {{abs \left({x}\right)}^{2}}}-{{x}^{3}}}\over{2 \ {{abs \left({x}\right)}^{4}}}$

(12)

Type: Expression(Integer)

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integrate(abs(x),x)

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

(13)

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

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

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

(14)

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

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

$\label{eq15}\int^{ \displaystyle x}{{{{{abs \left({\%A}\right)}^{2}}-{{\%A}^{2}}}\over{2 \ {{abs \left({\%A}\right)}^{3}}}}\ {d \%A}}$

(15)

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

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

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

(16)

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

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

$\label{eq17}\mbox{\tt "failed"}$

(17)

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/100)::DFLOAT for x in 1..50 by 5]

$\label{eq18}\left[{0.01}, \:{0.06}, \:{0.11}, \:{0.16}, \:{0.21}, \:{0.26}, \:{0.31}, \:{0.36}, \:{0.41}, \:{0.46}\right]$

(18)

Type: List(DoubleFloat?)

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

$\label{eq19}\begin{array}{@{}l} \displaystyle \left[{0.999850018747813}, \:{0.9946241983517127}, \: \right. \ \ \displaystyle \left.{0.9821206955196448}, \:{0.9627931278439013}, \: \right. \ \ \displaystyle \left.{0.9373177823982093}, \:{0.9065403772838817}, \: \right. \ \ \displaystyle \left.{0.8714144021297814}, \:{0.8329385728299635}, \: \right. \ \ \displaystyle \left.{0.792099910288845}, \:{0.7498270490909332}\right]$

(19)

Type: List(DoubleFloat?)

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gnuDraw(X,Y,"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) 
                              List(DoubleFloat)
                              List(DoubleFloat)
                                   String

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