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SandBoxDiracDelta

last edited 10 years ago by Bill page

Edit detail for SandBoxDiracDelta revision 39 of 40

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Editor: Bill page Time: 2014/08/19 19:33:47 GMT+0
Note: Expression Integer

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

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

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

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

changed:
-We are interested in the behaviour of these functions in the limit as
We are interested in the behavior of these functions in the limit as

removed:
-complexNormalize complexExpand %

changed:
-diracDelta(x^2)
-limit(%,b=0)
limit(diracDelta(x^2), b=0)

changed:
-unitDoublet(x^2)
-limit(%,b=0)
limit( unitDoublet(x^2) ,b=0)

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

changed:
-limit(d1,b=0)
-normalize(d1-diracDelta(x))
simplify(d1-diracDelta(x))

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)

fricas

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

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

(7)

Type: Boolean

fricas

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)

fricas

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

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

(10)

Type: Boolean

fricas

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

fricas

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)

fricas

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

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

(13)

Type: Boolean

fricas

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

fricas

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

fricas

limit( signum(1) ,b=0)

fricas

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

$\label{eq15}1$

(15)

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

fricas

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

fricas

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

$\label{eq16}- 1$

(16)

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

fricas

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)),...)

fricas

-- 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

fricas

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

$\label{eq24}0$

(24)

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

fricas

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

$\label{eq25}1$

(25)

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

fricas

f(x)==c1*x+c2

Type: Void

fricas

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)),...)

fricas

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

fricas

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

fricas

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

Type: Void

fricas

limit( heavisideStep(1), b=0)

   There are 12 exposed and 6 unexposed library operations named eval 
      having 2 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                              )display op eval
      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 eval 
      with argument type(s) 
            Union(Expression(Integer),List(Expression(Integer)))
                        Equation(Polynomial(Integer))

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.
   FriCAS will attempt to step through and interpret the code.

$\label{eq45}1$

(45)

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

fricas

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

$\label{eq46}0$

(46)

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

fricas

limit( heavisideStep(0) ,b=0)

$\label{eq47}1 \over 2$

(47)

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

fricas

limit( heavisideStep(x), x=0)

$\label{eq48}1 \over 2$

(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)

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

(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)

fricas

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)

   Cannot compile map: heavisideStep 
   We will attempt to interpret the code.

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

(69)

Type: Expression(Integer)

fricas

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

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

(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?)

fricas

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"