MathAction SandBoxSignum

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Derivative of abs

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s1:=D(abs(x),x)

$\label{eq1}{abs \left({x}\right)}\over x$

(1)

Type: Expression(Integer)

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D(s1,x)

$\label{eq2}0$

(2)

Type: Expression(Integer)

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limit(abs(x)/x,x=0,"right")

$\label{eq3}1$

(3)

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

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limit(abs(x)/x,x=0,"left")

$\label{eq4}- 1$

(4)

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

An approximation to signum

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D(tanh(a*x),x)

$\label{eq5}-{a \ {{\tanh \left({a \ x}\right)}^{2}}}+ a$

(5)

Type: Expression(Integer)

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limit(%,a=%plusInfinity)

$\label{eq6}\verb#"failed"#$

(6)

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

Patterns for derivatives

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Abs:=operator('Abs)

$\label{eq7}\hbox{\axiomType{Abs}\ }$

(7)

Type: BasicOperator?

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D(abs(x),x)

$\label{eq8}{abs \left({x}\right)}\over x$

(8)

Type: Expression(Integer)

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signum:=operator('signum)

$\label{eq9}signum$

(9)

Type: BasicOperator?

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diracDelta:=operator('diracDelta)

$\label{eq10}diracDelta$

(10)

Type: BasicOperator?

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signum(x)

$\label{eq11}signum \left({x}\right)$

(11)

Type: Expression(Integer)

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D(signum(x),[x,x])

$\label{eq12}{signum^{\prime \prime}}\left({x}\right)$

(12)

Type: Expression(Integer)

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diff1:=rule
  signum(x) == Abs(x)/x
  D(Abs(x),x) == signum(x) + 2*x*diracDelta(x)
  D(signum(x),x) == 2*diracDelta(x)

$\label{eq13}\begin{array}{@{}l} \displaystyle \left\{{= = \left({{signum \left({x}\right)}, \:{{\hbox{\axiomType{Abs}\ } \left({x}\right)}\over x}}\right)}, \: \right. \ \ \displaystyle \left.{= = \left({{{\hbox{\axiomType{Abs}\ }^{\prime}}\left({x}\right)}, \:{{signum \left({x}\right)}+{2 \ x \ {diracDelta \left({x}\right)}}}}\right)}, \: \right. \ \ \displaystyle \left.{= = \left({{{signum^{\prime}}\left({x}\right)}, \:{2 \ {diracDelta \left({x}\right)}}}\right)}\right\}$

(13)

Type: Ruleset(Integer,Integer,Expression(Integer))

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diff(x,y)==diff1 D(x,y)

Type: Void

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diff(Abs(x),x)

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

$\label{eq14}{{2 \ {{x}^{2}}\ {diracDelta \left({x}\right)}}+{\hbox{\axiomType{Abs}\ } \left({x}\right)}}\over x$

(14)

Type: Expression(Integer)

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diff(Abs(x)/x,x)

$\label{eq15}2 \ {diracDelta \left({x}\right)}$

(15)

Type: Expression(Integer)

Adding the singular part of the derivative

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diff(f,x) == D(f,x)+(limit(f,x=0,"right")-limit(f,x=0,"left"))::EXPR(INT)*diracDelta(x)

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

Type: Void

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diff(abs(x),x)

   There are 9 exposed and 6 unexposed library operations named - 
      having 2 argument(s) but none was determined to be applicable. 
      Use HyperDoc Browse, or issue
                                )display op -
      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 - 
      with argument type(s) 
           Union(OrderedCompletion(Expression(Integer)),"failed")
           Union(OrderedCompletion(Expression(Integer)),"failed")

      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.

$\label{eq16}{abs \left({x}\right)}\over x$

(16)

Type: Expression(Integer)

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diff(%,x)

$\label{eq17}2 \ {diracDelta \left({x}\right)}$

(17)

Type: Expression(Integer)

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signum(x)==abs(x)/x

Type: Void

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diff(signum(x),x)

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

$\label{eq18}2 \ {diracDelta \left({x}\right)}$

(18)

Type: Expression(Integer)

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integrate(signum(x),x)

$\label{eq19}\int^{ \displaystyle x}{{{abs \left({\%A}\right)}\over \%A}\ {d \%A}}$

(19)

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

Heaviside

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H(x)==(abs(x)+x)/2/x

Type: Void

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H(-0.3),H(0.3)

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Compiling function H with type Float -> Float

$\label{eq20}\left[{0.0}, \:{1.0}\right]$

(20)

Type: Tuple(Float)

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diff(H(x),x)

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

$\label{eq21}diracDelta \left({x}\right)$

(21)

Type: Expression(Integer)

abs and sqrt

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-- for x real
real:=operator('real)

$\label{eq22}real$

(22)

Type: BasicOperator?

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abs(real x)=sqrt(real(x)^2)

$\label{eq23}{abs \left({real \left({x}\right)}\right)}={\sqrt{{real \left({x}\right)}^{2}}}$

(23)

Type: Equation(Expression(Integer))

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-- OK
D(sqrt(x^2),x)

$\label{eq24}x \over{\sqrt{{x}^{2}}}$

(24)

Type: Expression(Integer)

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integrate(%,x)

$\label{eq25}\sqrt{{x}^{2}}$

(25)

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

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-- Wrong
integrate(sqrt(x^2),x)

$\label{eq26}{{x}^{2}}\over 2$

(26)

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

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D(%,x)

$\label{eq27}x$

(27)

Type: Expression(Integer)

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-- Expected
x*sqrt(x^2)/2

$\label{eq28}{x \ {\sqrt{{x}^{2}}}}\over 2$

(28)

Type: Expression(Integer)

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D(%,x)

$\label{eq29}{{x}^{2}}\over{\sqrt{{x}^{2}}}$

(29)

Type: Expression(Integer)

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integrate(sqrt(x^2),x=-2..2,"noPole")

$\label{eq30}0$

(30)

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

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integrate(x,x=-2..2)

$\label{eq31}0$

(31)

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

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integrate(abs(x),x=-2..2,"noPole")

$\label{eq32}\verb#"failed"#$

(32)

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

Solving differential equations, e.g.

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-- differential Operator
L(f)==D(f,[x,x])

Type: Void

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-- Green function
g(x)==abs(x)/2

Type: Void

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-- pulse
c(x)==H(x+1/2)-H(x-1/2)

Type: Void

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c(x)

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

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

$\label{eq33}{{{\left({2 \ x}- 1 \right)}\ {abs \left({{{2 \ x}+ 1}\over 2}\right)}}+{{\left(-{2 \ x}- 1 \right)}\ {abs \left({{{2 \ x}- 1}\over 2}\right)}}}\over{{4 \ {{x}^{2}}}- 1}$

(33)

Type: Expression(Integer)

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-- convolution
integrand:=c(x-y)*g(y)

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

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

$\label{eq34}{\left({\left({ \begin{array}{@{}l} \displaystyle {{\left({2 \ y}-{2 \ x}- 1 \right)}\ {abs \left({{{2 \ y}-{2 \ x}+ 1}\over 2}\right)}}+ \ \ \displaystyle {{\left(-{2 \ y}+{2 \ x}- 1 \right)}\ {abs \left({{{2 \ y}-{2 \ x}- 1}\over 2}\right)}}$

(34)

Type: Expression(Integer)

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conv:=integrate(integrand,y=%minusInfinity..%plusInfinity)

$\label{eq35}\verb#"potentialPole"#$

(35)

Type: Union(pole: potentialPole,...)

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conv:=integrate(integrand,y=%minusInfinity..%plusInfinity,"noPole")

$\label{eq36}\verb#"failed"#$

(36)

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

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conv:=integrate(integrand,y)

$\label{eq37}\int^{ \displaystyle y}{{{{{\left({2 \ x}-{2 \ \%A}- 1 \right)}\ {abs \left({\%A}\right)}\ {abs \left({{{2 \ x}-{2 \ \%A}+ 1}\over 2}\right)}}+{{\left(-{2 \ x}+{2 \ \%A}- 1 \right)}\ {abs \left({\%A}\right)}\ {abs \left({{{2 \ x}-{2 \ \%A}- 1}\over 2}\right)}}}\over{{8 \ {{x}^{2}}}-{{16}\ \%A \ x}+{8 \ {{\%A}^{2}}}- 2}}\ {d \%A}}$

(37)

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

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-- expected result
conv:=1/2*x^2*H(x+1/2)-1/2*x+1/2*x*H(x+1/2)+1/8*H(x+1/2)-1/2*x^2*H(x-1/2)+1/2*x*H(x-1/2)-1/8*H(x-1/2)

$\label{eq38}{{{\left({2 \ x}+ 1 \right)}\ {abs \left({{{2 \ x}+ 1}\over 2}\right)}}+{{\left(-{2 \ x}+ 1 \right)}\ {abs \left({{{2 \ x}- 1}\over 2}\right)}}}\over 8$

(38)

Type: Expression(Integer)

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-- ramp
diff(conv,x)

$\label{eq39}{{abs \left({{{2 \ x}+ 1}\over 2}\right)}-{abs \left({{{2 \ x}- 1}\over 2}\right)}}\over 2$

(39)

Type: Expression(Integer)

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test(L conv = c(x))

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

$\label{eq40} \mbox{\rm true}$

(40)

Type: Boolean

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gnuDraw(H(x), x=-2..2, "SandBoxSignum1.dat")

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Compiling function %R with type DoubleFloat -> DoubleFloat

Type: Void

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gnuDraw(c(x), x=-2..2, "SandBoxSignum2.dat")

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Compiling function %S with type DoubleFloat -> DoubleFloat

Type: Void

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gnuDraw(conv, x=-3..3, "SandBoxSignum3.dat")

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Compiling function %T with type DoubleFloat -> DoubleFloat

Type: Void

[terminal=pslatex,scale=1.3]
set yrange [0:2]
load "SandBoxSignum3.dat"