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SandBoxDiracDelta

last edited 10 years ago by Bill page

Edit detail for SandBoxDiracDelta revision 40 of 40

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Editor: Bill page Time: 2014/08/19 19:35:15 GMT+0
Note: Expression Integer

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

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

fricas

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

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

fricas

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

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

(1)

Type: Expression(Integer)

fricas

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.

fricas

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)

fricas

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

fricas

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

fricas

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)

fricas

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)

fricas

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 .

fricas

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)

fricas

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

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

(8)

Type: Expression(Integer)

fricas

--
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)

fricas

--
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)),...)

fricas

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)

fricas

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

$\label{eq17}0$

(17)

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

fricas

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

$\label{eq18}1$

(18)

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

fricas

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

fricas

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

fricas

limit( diracDelta(1) ,b=0)

fricas

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

$\label{eq20}0$

(20)

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

fricas

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

fricas

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

$\label{eq21}0$

(21)

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

fricas

limit( diracDelta(0) ,b=0)

fricas

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) )

fricas

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))

fricas

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

fricas

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

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

(26)

Type: Boolean

fricas

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)

fricas

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

$\label{eq29}1 \over 2$

(29)

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

fricas

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

$\label{eq30}0$

(30)

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

fricas

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

fricas

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)

fricas

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

$\label{eq33}0$

(33)

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

fricas

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

fricas

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

$\label{eq34}0$

(34)

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

fricas

limit( unitDoublet(0) ,b=0)

fricas

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

$\label{eq35}0$

(35)

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

fricas

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

fricas

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

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

(36)

Type: Boolean

fricas

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

$\label{eq37}0$

(37)

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

fricas

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

$\label{eq38}0$

(38)

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

fricas

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

fricas

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

fricas

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

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

(39)

Type: Boolean

fricas

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

$\label{eq40}0$

(40)

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

fricas

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

$\label{eq41}0$

(41)

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

fricas

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

$\label{eq42}0$

(42)

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

fricas

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

$\label{eq43}0$

(43)

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

Heaviside

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

fricas

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

fricas

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)

fricas

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

Type: Void

fricas

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)),...)

fricas

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)),...)

fricas

limit( heavisideStep(0) ,b=0)

fricas

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

$\label{eq47}1 \over 2$

(47)

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

fricas

limit( heavisideStep(x), x=0)

fricas

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

$\label{eq48}1 \over 2$

(48)

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

fricas

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

$\label{eq49}1$

(49)

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

fricas

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

$\label{eq50}0$

(50)

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

fricas

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

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

(51)

Type: Boolean

fricas

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

fricas

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

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

(52)

Type: Boolean

fricas

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

fricas

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)),...)

fricas

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?

fricas

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

$\label{eq55}1 \over c$

(55)

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

fricas

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

fricas

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

fricas

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

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

(56)

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

fricas

-- 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)

fricas

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

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

(58)

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

fricas

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

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

(59)

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

fricas

-- 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),...)

fricas

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),...)

fricas

limit(%,b=0)

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

(62)

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

fricas

-- 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),...)

fricas

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),...)

fricas

limit(%,b=0)

$\label{eq65}x$

(65)

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

fricas

-- 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),...)

fricas

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

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

(67)

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

Convolution

fricas

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

fricas

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

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

(68)

Type: Expression(Integer)

fricas

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

fricas

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

fricas

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)

fricas

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

fricas

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

fricas

)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];

fricas

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

Type: List(DoubleFloat?)

fricas

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"