MathAction SandBoxDiracDelta

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        SandBoxDiracDelta <-- You are here.

Distributions

We represent distributions as a complex function in the limit where the imaginary part goes to 0. Signum (sign), delta and doublet functions arise from the derivative of the absolute value function with respect to its real part.

Ref.

Derivatives of Distributions (Wikipedia)

Use definition of conjugate and derivative of abs(x) from SandBoxFunctionalSpecialFunction

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
wirtingerD(ex,z) == eval(D(eval(ex,z=%conjugate),%conjugate),%conjugate=z)

Type: Void

Absolute value as a function of real or complex variables

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abs(z:Expression Integer):Expression Integer == sqrt(z*conjugate(z)); abs(z)

   Function declaration abs : Expression(Integer) -> Expression(Integer
      ) has been added to workspace.

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Compiling function abs with type Expression(Integer) -> Expression(
      Integer)

$\label{eq1}\sqrt{z \ {\overline z}}$

(1)

Type: Expression(Integer)

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abz(a:Expression Integer):Expression Integer == sqrt((a+%i*b)*(a-%i*b)); abz(x)

   Function declaration abz : Expression(Integer) -> Expression(Integer
      ) has been added to workspace.

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Compiling function abz with type Expression(Integer) -> Expression(
      Integer)

$\label{eq2}\sqrt{{{x}^{2}}+{{b}^{2}}}$

(2)

Type: Expression(Integer)

Total Wirtinger derivatives of this complex function (CR-calculus)

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abs':=wirtingerD(abs(z),z)+wirtingerD(abs(z),conjugate z)

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

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Compiling function wirtingerD with type (Expression(Integer),
      Expression(Integer)) -> Expression(Integer)

$\label{eq3}{{\overline z}+ z}\over{2 \ {\sqrt{z \ {\overline z}}}}$

(3)

Type: Expression(Integer)

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abs'':=wirtingerD(abs',z)+wirtingerD(abs',conjugate z)

$\label{eq4}{-{{\overline z}^{2}}+{2 \ z \ {\overline z}}-{{z}^{2}}}\over{4 \ z \ {\overline z}\ {\sqrt{z \ {\overline z}}}}$

(4)

Type: Expression(Integer)

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abs''':=wirtingerD(abs'',z)+wirtingerD(abs'',conjugate z)

$\label{eq5}{{3 \ {{\overline z}^{3}}}-{3 \ z \ {{\overline z}^{2}}}-{3 \ {{z}^{2}}\ {\overline z}}+{3 \ {{z}^{3}}}}\over{8 \ {{z}^{2}}\ {{\overline z}^{2}}\ {\sqrt{z \ {\overline z}}}}$

(5)

Type: Expression(Integer)

Consider these as complex functions of for real and .

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realSignum:Expression Integer:=eval(eval(abs',z=a+%i*b),[conjugate(a)=a,conjugate(b)=b])

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

(6)

Type: Expression(Integer)

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test(realSignum=D(abz(a),a))

$\label{eq7} \mbox{\rm true}$

(7)

Type: Boolean

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signum(x)==eval(realSignum,a=x); signum(x)

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

$\label{eq8}x \over{\sqrt{{{x}^{2}}+{{b}^{2}}}}$

(8)

Type: Expression(Integer)

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--
realDirac:Expression Integer:=eval(eval(abs'',z=a+%i*b),[conjugate(a)=a,conjugate(b)=b])

$\label{eq9}{{b}^{2}}\over{{\left({{b}^{2}}+{{a}^{2}}\right)}\ {\sqrt{{{b}^{2}}+{{a}^{2}}}}}$

(9)

Type: Expression(Integer)

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test(realDirac=D(abz(a),[a,a]))

$\label{eq10} \mbox{\rm true}$

(10)

Type: Boolean

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diracDelta(x)==eval(realDirac/2,a=x); diracDelta(x)

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

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

(11)

Type: Expression(Integer)

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--
realUnitDoublet:Expression Integer:=eval(eval(abs''',z=a+%i*b),[conjugate(a)=a,conjugate(b)=b])

$\label{eq12}-{{3 \ a \ {{b}^{2}}}\over{{\left({{b}^{4}}+{2 \ {{a}^{2}}\ {{b}^{2}}}+{{a}^{4}}\right)}\ {\sqrt{{{b}^{2}}+{{a}^{2}}}}}}$

(12)

Type: Expression(Integer)

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test(realUnitDoublet=D(abz(a),[a,a,a]))

$\label{eq13} \mbox{\rm true}$

(13)

Type: Boolean

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unitDoublet(x)==eval(-realUnitDoublet/2,a=x); unitDoublet(x)

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

$\label{eq14}{3 \ {{b}^{2}}\ x}\over{{\left({2 \ {{x}^{4}}}+{4 \ {{b}^{2}}\ {{x}^{2}}}+{2 \ {{b}^{4}}}\right)}\ {\sqrt{{{x}^{2}}+{{b}^{2}}}}}$

(14)

Type: Expression(Integer)

We are interested in the behavior of these functions in the limit as $b \to 0$ .

signum Properties

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limit( signum(1) ,b=0)

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

$\label{eq15}1$

(15)

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

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limit( signum(-1) ,b=0)

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

$\label{eq16}- 1$

(16)

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

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limit( signum(0) ,b=0)

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

$\label{eq17}0$

(17)

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

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limit( signum(x)^2 ,b=0)

$\label{eq18}1$

(18)

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

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-- odd function
test( signum(x)=-signum(-x))

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

$\label{eq19} \mbox{\rm true}$

(19)

Type: Boolean

diracDelta Properties

http://en.wikipedia.org/wiki/Dirac_delta_function

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limit( diracDelta(1) ,b=0)

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

$\label{eq20}0$

(20)

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

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limit( diracDelta(-1) ,b=0)

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

$\label{eq21}0$

(21)

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

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limit( diracDelta(0) ,b=0)

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

$\label{eq22}+ \infty$

(22)

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

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-- even function
test( diracDelta(x)=diracDelta(-x) )

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

$\label{eq23} \mbox{\rm true}$

(23)

Type: Boolean

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limit( diracDelta(x)^2 ,b=0)

$\label{eq24}0$

(24)

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

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

$\label{eq25}1$

(25)

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

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f(x)==c1*x+c2

Type: Void

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test(integrate(f(x)*diracDelta(x-t),x=%minusInfinity..%plusInfinity,"noPole")=f(t))

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

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

$\label{eq26} \mbox{\rm true}$

(26)

Type: Boolean

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limit(integrate(log(x)*diracDelta(x-t),x=%minusInfinity..%plusInfinity,"noPole"), b=0)

$\label{eq27}{\log \left({{t}^{2}}\right)}\over 2$

(27)

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

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eval(%,t=exp(x))

$\label{eq28}x$

(28)

Type: Expression(Integer)

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integrate(diracDelta(2*x),x=%minusInfinity..%plusInfinity,"noPole")

$\label{eq29}1 \over 2$

(29)

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

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--
limit( signum(x)*diracDelta(x) ,b=0)

$\label{eq30}0$

(30)

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

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limit(diracDelta(x^2), b=0)

$\label{eq31}0$

(31)

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

The Unit Doublet function comes after diracDelta

http://en.wikipedia.org/wiki/Unit_doublet

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test(unitDoublet(x)=3*diracDelta(x)*signum(x)/abz(x))

$\label{eq32} \mbox{\rm true}$

(32)

Type: Boolean

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limit( unitDoublet(1) ,b=0)

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Compiling function unitDoublet with type PositiveInteger -> 
      Expression(Integer)

$\label{eq33}0$

(33)

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

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limit( unitDoublet(-1) ,b=0)

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Compiling function unitDoublet with type Integer -> Expression(
      Integer)

$\label{eq34}0$

(34)

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

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limit( unitDoublet(0) ,b=0)

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Compiling function unitDoublet with type NonNegativeInteger -> 
      Expression(Integer)

$\label{eq35}0$

(35)

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

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-- odd function
test( unitDoublet(x)=-unitDoublet(-x) )

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

$\label{eq36} \mbox{\rm true}$

(36)

Type: Boolean

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limit( unitDoublet(x)^2 ,b=0)

$\label{eq37}0$

(37)

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

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

$\label{eq38}0$

(38)

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

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f(x)==c1*x^2+c2*x+c3

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

Type: Void

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test(integrate(f(x)*unitDoublet(x-t),x=%minusInfinity..%plusInfinity,"noPole")=eval(D(f(x),x),x=t))

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

$\label{eq39} \mbox{\rm true}$

(39)

Type: Boolean

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integrate(unitDoublet(2*x),x=%minusInfinity..%plusInfinity,"noPole")

$\label{eq40}0$

(40)

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

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--
limit( signum(x)*unitDoublet(x) ,b=0)

$\label{eq41}0$

(41)

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

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limit( diracDelta(x)*unitDoublet(x) ,b=0)

$\label{eq42}0$

(42)

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

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limit( unitDoublet(x^2) ,b=0)

$\label{eq43}0$

(43)

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

Heaviside

http://en.wikipedia.org/wiki/Unit_step_function

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realHeaviside:Expression Integer := integrate(diracDelta(a),a)

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

$\label{eq44}-{{{b}^{2}}\over{{2 \ a \ {\sqrt{{{b}^{2}}+{{a}^{2}}}}}-{2 \ {{b}^{2}}}-{2 \ {{a}^{2}}}}}$

(44)

Type: Expression(Integer)

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heavisideStep(x)==eval(realHeaviside,a=x)

Type: Void

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limit( heavisideStep(1), b=0)

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Compiling function heavisideStep with type PositiveInteger -> 
      Expression(Integer)

$\label{eq45}1$

(45)

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

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limit( heavisideStep(-1) ,b=0)

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Compiling function heavisideStep with type Integer -> Expression(
      Integer)

$\label{eq46}0$

(46)

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

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limit( heavisideStep(0) ,b=0)

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Compiling function heavisideStep with type NonNegativeInteger -> 
      Expression(Integer)

$\label{eq47}1 \over 2$

(47)

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

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limit( heavisideStep(x), x=0)

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

$\label{eq48}1 \over 2$

(48)

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

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limit( heavisideStep(x) ,x=%plusInfinity)

$\label{eq49}1$

(49)

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

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limit( heavisideStep(x) ,x=%minusInfinity)

$\label{eq50}0$

(50)

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

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test(simplify( (1+signum(x))/2 - heavisideStep(x) ) = 0)

$\label{eq51} \mbox{\rm true}$

(51)

Type: Boolean

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test(limit( integrate(heavisideStep(t),t=%minusInfinity..x, "noPole") - x*heavisideStep(x), b=0) =0)

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Compiling function heavisideStep with type Variable(t) -> Expression
      (Integer)

$\label{eq52} \mbox{\rm true}$

(52)

Type: Boolean

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-- diracDelta?
d1:=integrate(heavisideStep(t)*unitDoublet(t),t=%minusInfinity..x, "noPole")

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Compiling function unitDoublet with type Variable(t) -> Expression(
      Integer)

$\label{eq53}{\left( \begin{array}{@{}l} \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle {{24}\ {{x}^{7}}}+ \ \ \displaystyle {{66}\ {{b}^{2}}\ {{x}^{5}}}+ \ \ \displaystyle {{60}\ {{b}^{4}}\ {{x}^{3}}}+ \ \ \displaystyle {{18}\ {{b}^{6}}\ x}$

(53)

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

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simplify(d1-diracDelta(x))

$\label{eq54}{\left( \begin{array}{@{}l} \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle {{24}\ {{x}^{6}}}+ \ \ \displaystyle {{54}\ {{b}^{2}}\ {{x}^{4}}}+ \ \ \displaystyle {{36}\ {{b}^{4}}\ {{x}^{2}}}+{6 \ {{b}^{6}}}$

(54)

Type: Expression(Integer)

Problems?

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-- expected 1/abs(c)
integrate(diracDelta(c*x),x=%minusInfinity..%plusInfinity,"noPole")

$\label{eq55}1 \over c$

(55)

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

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f(x)==c1*x^2+c2*x+c3

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

Type: Void

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

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

$\label{eq56}\mbox{\tt "failed"}$

(56)

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

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-- expected
f(t)

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

$\label{eq57}{c 1 \ {{t}^{2}}}+{c 2 \ t}+ c 3$

(57)

Type: Polynomial(Integer)

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-- expected 1
integrate(exp(x)*diracDelta(x),x=%minusInfinity..%plusInfinity,"noPole")

$\label{eq58}\mbox{\tt "failed"}$

(58)

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

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-- expected 1
integrate(exp(x)*unitDoublet(x),x=%minusInfinity..%plusInfinity,"noPole")

$\label{eq59}\mbox{\tt "failed"}$

(59)

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

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-- loops forever on %plusInfinity limit of indefinite integral
--integrate(log(x)*unitDoublet(x-t),x=%minusInfinity..%plusInfinity,"noPole")
-- expected x*abs(x)/2
integrate(abs(x),x)

$\label{eq60}\int^{ \displaystyle x}{{\sqrt{\%A \ {\overline \%A}}}\ {d \%A}}$

(60)

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

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integrate(abz(x),x)

$\label{eq61}{\left( \begin{array}{@{}l} \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle -{2 \ {{b}^{2}}\ x \ {\sqrt{{{x}^{2}}+{{b}^{2}}}}}+ \ \ \displaystyle {2 \ {{b}^{2}}\ {{x}^{2}}}+{{b}^{4}}$

(61)

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

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limit(%,b=0)

$\label{eq62}{{x}^{2}}\over 2$

(62)

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

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-- expected abs(x)
integrate(abs',z)

$\label{eq63}\int^{ \displaystyle z}{{{{\overline \%A}+ \%A}\over{2 \ {\sqrt{\%A \ {\overline \%A}}}}}\ {d \%A}}$

(63)

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

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integrate(signum(x),x)

$\label{eq64}{-{x \ {\sqrt{{{x}^{2}}+{{b}^{2}}}}}+{{x}^{2}}+{{b}^{2}}}\over{{\sqrt{{{x}^{2}}+{{b}^{2}}}}- x}$

(64)

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

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limit(%,b=0)

$\label{eq65}x$

(65)

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

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-- expected signum(x)/2
integrate(abs'',z)

$\label{eq66}\int^{ \displaystyle z}{{{-{{\overline \%A}^{2}}+{2 \ \%A \ {\overline \%A}}-{{\%A}^{2}}}\over{4 \ \%A \ {\overline \%A}\ {\sqrt{\%A \ {\overline \%A}}}}}\ {d \%A}}$

(66)

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

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integrate(diracDelta(x^2+y^2), x=%minusInfinity..%plusInfinity, "noPole")

$\label{eq67}\mbox{\tt "failed"}$

(67)

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

Convolution

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g(x)==abz(x)/2; g(y)

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Compiling function g with type Variable(y) -> Expression(Integer)

$\label{eq68}{\sqrt{{{y}^{2}}+{{b}^{2}}}}\over 2$

(68)

Type: Expression(Integer)

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h(x)==heavisideStep(x+1/2)-heavisideStep(x-1/2); h(x)

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Compiling function heavisideStep with type Polynomial(Fraction(
      Integer)) -> Expression(Integer)

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

$\label{eq69}{\left( \begin{array}{@{}l} \displaystyle {{\left({4 \ {{b}^{2}}\ x}+{2 \ {{b}^{2}}}\right)}\ {\sqrt{{4 \ {{x}^{2}}}+{4 \ x}+{4 \ {{b}^{2}}}+ 1}}}+ \ \ \displaystyle {{\left(-{4 \ {{b}^{2}}\ x}+{2 \ {{b}^{2}}}\right)}\ {\sqrt{{4 \ {{x}^{2}}}-{4 \ x}+{4 \ {{b}^{2}}}+ 1}}}-{{16}\ {{b}^{2}}\ x}$

(69)

Type: Expression(Integer)

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conv:=integrate(h(x-y)*g(y),y=%minusInfinity..%plusInfinity,"noPole")

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

$\label{eq70}\mbox{\tt "failed"}$

(70)

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

Representing a distribution as a series of bump functions

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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 1];

Type: List(DoubleFloat?)

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Y:=[eval(unitDoublet(x),b=1.0)::DFLOAT for x in X];

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Compiling function unitDoublet with type DoubleFloat -> Expression(
      DoubleFloat)

Type: List(DoubleFloat?)

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gnuDraw(X,Y,"SandBoxDiracDeltaX2.dat")

   Graph data being transmitted to the viewport manager...
   FriCAS2D data being transmitted to the viewport manager...

Type: Void

[terminal=pslatex,terminaloptions=color,scale=1.3]
load "SandBoxDiracDeltaX2.dat"