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SandBoxDiracDelta

last edited 9 years ago by Bill page

Edit detail for SandBoxDiracDelta revision 35 of 40

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Editor: Bill Page Time: 2014/08/19 01:25:54 GMT+0
Note: Unit Doublet

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

added:
- "Derivatives of Distributions":http://en.wikipedia.org/wiki/Distribution_(mathematics)#Derivatives_of_distributions
  Wikipedia

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

fricas

abs(z)==sqrt(z*conjugate(z)); abs(z)

fricas

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

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

(1)

Type: Expression(Integer)

Or as a function of real variables

fricas

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

fricas

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

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

(2)

Type: Expression(Complex(Integer))

Total Wirtinger derivative of complex variables (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 a complex function of for real and with $b \to 0$ .

fricas

realSignum:=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(Complex(Integer))

fricas

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

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

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

(7)

Type: Boolean

fricas

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

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

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

(8)

Type: Expression(Complex(Integer))

fricas

--
realDirac:=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(Complex(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(
      Complex(Integer))

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

(11)

Type: Expression(Complex(Integer))

fricas

--
realUnitDoublet:=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(Complex(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(
      Complex(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(Complex(Integer))

signum Properties

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limit( signum(1) ,b=0)

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

$\label{eq15}1$

(15)

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

fricas

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

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

$\label{eq16}- 1$

(16)

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

fricas

limit( signum(0) ,b=0)

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

$\label{eq17}0$

(17)

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

fricas

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

$\label{eq18}1$

(18)

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

fricas

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

fricas

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

$\label{eq20}0$

(20)

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

fricas

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

fricas

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

$\label{eq21}0$

(21)

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

fricas

limit( diracDelta(0) ,b=0)

fricas

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

$\label{eq22}+ \infty$

(22)

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

fricas

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

fricas

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

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

(23)

Type: Boolean

fricas

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

$\label{eq24}0$

(24)

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

fricas

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

$\label{eq25}1$

(25)

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

fricas

f(x)==x+c

Type: Void

fricas

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

fricas

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

$\label{eq26}c$

(26)

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

fricas

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

$\label{eq27}t + c$

(27)

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

fricas

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

$\label{eq28}1 \over 2$

(28)

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

fricas

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

$\label{eq29}\mbox{\tt "failed"}$

(29)

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

fricas

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

$\label{eq30}{{\log \left({{{b}^{4}}\over 4}\right)}-{\log \left({4}\right)}}\over 4$

(30)

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

fricas

limit( %, b=0)

$\label{eq31}- \infty$

(31)

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

fricas

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

$\label{eq32}0$

(32)

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

fricas

diracDelta(x^2)

$\label{eq33}{{b}^{2}}\over{{\left({2 \ {{x}^{4}}}+{2 \ {{b}^{2}}}\right)}\ {\sqrt{{{x}^{4}}+{{b}^{2}}}}}$

(33)

Type: Expression(Complex(Integer))

fricas

limit(%,b=0)

$\label{eq34}0$

(34)

Type: Union(OrderedCompletion?(Expression(Complex(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))

$\label{eq35} \mbox{\rm true}$

(35)

Type: Boolean

fricas

limit( unitDoublet(1) ,b=0)

fricas

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

$\label{eq36}0$

(36)

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

fricas

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

fricas

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

$\label{eq37}0$

(37)

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

fricas

limit( unitDoublet(0) ,b=0)

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

$\label{eq38}0$

(38)

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

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

fricas

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

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

(39)

Type: Boolean

fricas

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

$\label{eq40}0$

(40)

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

fricas

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

$\label{eq41}0$

(41)

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

fricas

f(x)==x+c

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

Type: Void

fricas

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

fricas

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

$\label{eq42}1$

(42)

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

fricas

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

$\label{eq43}1$

(43)

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

fricas

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

$\label{eq44}0$

(44)

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

fricas

f(x)==c*x

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

Type: Void

fricas

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

fricas

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

$\label{eq45}c$

(45)

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

fricas

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

$\label{eq46}0$

(46)

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

fricas

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

$\label{eq47}0$

(47)

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

fricas

unitDoublet(x^2)

$\label{eq48}{3 \ {{b}^{2}}\ {{x}^{2}}}\over{{\left({2 \ {{x}^{8}}}+{4 \ {{b}^{2}}\ {{x}^{4}}}+{2 \ {{b}^{4}}}\right)}\ {\sqrt{{{x}^{4}}+{{b}^{2}}}}}$

(48)

Type: Expression(Complex(Integer))

fricas

limit(%,b=0)

$\label{eq49}0$

(49)

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

Heaviside

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

fricas

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

fricas

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

$\label{eq50}-{{{b}^{2}}\over{{2 \ a \ {\sqrt{{{b}^{2}}+{{a}^{2}}}}}-{2 \ {{b}^{2}}}-{2 \ {{a}^{2}}}}}$

(50)

Type: Expression(Complex(Integer))

fricas

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

Type: Void

fricas

limit( heavisideStep(1), b=0)

fricas

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

$\label{eq51}1$

(51)

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

fricas

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

fricas

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

$\label{eq52}0$

(52)

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

fricas

limit( heavisideStep(0) ,b=0)

fricas

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

$\label{eq53}1 \over 2$

(53)

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

fricas

limit( heavisideStep(x), x=0)

fricas

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

$\label{eq54}1 \over 2$

(54)

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

fricas

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

$\label{eq55}1$

(55)

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

fricas

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

$\label{eq56}0$

(56)

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

fricas

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

$\label{eq57} \mbox{\rm true}$

(57)

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
      (Complex(Integer))

$\label{eq58} \mbox{\rm true}$

(58)

Type: Boolean

Problems?

fricas

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

$\label{eq59}0$

(59)

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

fricas

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

$\label{eq60}\mbox{\tt "failed"}$

(60)

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

fricas

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

$\label{eq61}0$

(61)

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

fricas

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

fricas

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

$\label{eq62}\int^{ \displaystyle x}{{\sqrt{\%A \ {\overline \%A}}}\ {d \%A}}$

(62)

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

fricas

integrate(abz(x),x)

$\label{eq63}{\left( \begin{array}{@{}l} \displaystyle {{\left({ \begin{array}{@{}l} \displaystyle -{2 \ {{b}^{2}}\ x \ {\sqrt{{{x}^{2}}+{{b}^{2}}}}}+ \ \ \displaystyle {2 \ {{b}^{2}}\ {{x}^{2}}}+{{b}^{4}}$

(63)

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

fricas

limit(%,b=0)

$\label{eq64}-{{{x}^{2}}\over 2}$

(64)

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

fricas

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

$\label{eq65}{{\%z \ {\overline z}}+{\%z \ z}}\over{2 \ {\sqrt{z \ {\overline z}}}}$

(65)

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

fricas

integrate(signum(x),x)

$\label{eq66}{-{x \ {\sqrt{{{x}^{2}}+{{b}^{2}}}}}+{{x}^{2}}+{{b}^{2}}}\over{{\sqrt{{{x}^{2}}+{{b}^{2}}}}- x}$

(66)

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

fricas

limit(%,b=0)

$\label{eq67}- x$

(67)

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

fricas

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

$\label{eq68}{-{\%z \ {{\overline z}^{2}}}+{2 \ \%z \ z \ {\overline z}}-{\%z \ {{z}^{2}}}}\over{4 \ z \ {\overline z}\ {\sqrt{z \ {\overline z}}}}$

(68)

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

fricas

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

$\label{eq69}\mbox{\tt "failed"}$

(69)

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

Convolution

fricas

g(x)==eval((abs(a+%i*b)/2+abs(a-%i*b)/2),a=x)/2

Type: Void

fricas

g(y)

fricas

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

fricas

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

$\label{eq70}{{\sqrt{{{\left(y +{i \ b}\right)}\ {\overline y}}+{{\left(-{i \ y}+ b \right)}\ {\overline b}}}}+{\sqrt{{{\left(y -{i \ b}\right)}\ {\overline y}}+{{\left({i \ y}+ b \right)}\ {\overline b}}}}}\over 4$

(70)

Type: Expression(Complex(Integer))

fricas

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

Type: Void

fricas

h(x)

fricas

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

fricas

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

$\label{eq71}{\left( \begin{array}{@{}l} \displaystyle {{\left({4 \ {{b}^{2}}\ x}+{2 \ {{b}^{2}}}\right)}\ {\sqrt{{4 \ {{x}^{2}}}+{4 \ x}+{4 \ {{b}^{2}}}+ 1}}}+ \ \ \displaystyle {{\left(-{4 \ {{b}^{2}}\ x}+{2 \ {{b}^{2}}}\right)}\ {\sqrt{{4 \ {{x}^{2}}}-{4 \ x}+{4 \ {{b}^{2}}}+ 1}}}-{{16}\ {{b}^{2}}\ x}$

(71)

Type: Expression(Complex(Integer))

fricas

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

fricas

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

$\label{eq72}\mbox{\tt "failed"}$

(72)

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?)

fricas

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

fricas

Compiling function unitDoublet with type DoubleFloat -> Expression(
      Complex(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"