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Examples of Fractional Calculus Computer Algebra System


Handbook

Arithmetic >>

Exact computation

  • Fraction `1E2-1/2`
  • Big number: add prefix "big" to number big1234567890123456789

  • Complex

  • Convert complex a+b*i to polar(r,theta) coordinates

  • convert 3-4i to polar
    topolar(3-4i)
  • input complex number in polar(r,theta) coordinates
    polar(3,45degree)

  • Convert to complex
    tocomplex(polar(3,45degree))

  • input complex number in cis(theta) format
    cis(45degree)
  • Convert back by numeric computation
    n(cis(45degree))

  • Numerical approximations

  • n( sin(pi/4) )
  • n( sin(30 degree) )

  • `sin^(0.5)(1)` is the 0.5 order derivative of sin(x) at x=1
    n( sin(0.5,1) )
  • `sin(1)^(0.5)` is the 0.5 power of sin(x) at x=1
    n( sin(1)^0.5 )
  • Algebra >>

  • simplify
    simplify( (x^2 - 1)/(x-1) )
  • expand
    expand( (x-1)^3 )
  • factor
    Factorization
    factor( x^4-1 )

  • factorizing
    factor( x^2+3*x+2 )
  • tangent at x=1
    tangent( sin(x),x=1 )

    Convert

    convert to power
  • topower( cos(x) )

  • convert to trig
  • convert exp(x) to trig

  • convert sin(x) to exp(x),
  • convert sin(x) to exp
  • toexp( sin(x) )

    Convert to exp(x)

  • toexp(gamma(2,x))
  • inverse
    inverse( sin(x) )

    polymonial:

    topoly convert polymonial to polys() as holder of polymonial coefficients,

  • convert `x^2-5*x+6` to poly
  • topoly( `x^2-5*x+6` )
    activate polys() to polymonial
  • simplify( polys(1,-5,6,x) )

    topolyroot convert a polymonial to polyroots() as holder of polymonial roots,

  • convert (x^2-1) to polyroot
  • topolyroot(x^2-1)
    activate polyroots() to polymonial
  • simplify( polyroots(2,3,x) )

  • Calculus >>

    Limit

    lim sin(x)/x as x->0
  • lim sin(x)/x as x->0
  • `lim _(x->oo) log(x)/x`

  • lim( log(x)/x as x->inf )
    by default x=infinity,
  • lim(log(x)/x)

    Derivatives

    Differentiate
  • `d/dx sin(x)`
  • d(sin(x))

    Second order derivative

  • `d^2/dx^2 sin(x)`
  • d(sin(x),x,2)

    sin(0.5,x) is inert holder of the 0.5 order derivative `sin^((0.5))(x)`, it can be activated by activate() or simplify():

  • activate( sin(0.5,x) )
  • `d^(0.5)/dx^(0.5) sin(x)`
  • d(sin(x),x,0.5)

    semiderivative

  • `d^(0.5)/dx^(0.5) sin(x)`
  • semid(sin(x))

    Derivative as x=1

  • `d/dx | _(x=1) x^6`
  • d( x^6 as x=1 )

    Second order derivative as x=1

  • `d^2/dx^2 | _(x=1) x^6`
  • d(x^6 as x=1,2)

    Fractional calculus

  • input sin(0.5,x) as the 0.5 order derivative of sin(x) for `sin^((0.5))(x)`
    `sin^((0.5))(x)`
  • simplify sin(0.5,x) as the 0.5 order derivative of sin(x) for `sin^((0.5))(x)`
    simplify `sin^((0.5))(x)`
  • 0.5 order derivative, semiderivative, semid()
    `d^0.5/dx^0.5 log(x)`
  • 0.5 order derivative again
    `d^(0.5)/dx^(0.5) d^(0.5)/dx^(0.5) sin(x)`
  • Minus order derivative
    `d^(-0.5)/dx^(-0.5) sin(x)`
  • inverse the 0.5 order derivative of sin(x) function
    (-1)( sin(0.5)(x) )
  • Derive the product rule
    `d/dx` (f(x)*g(x)*h(x))
  • …as well as the quotient rule
    `d/dx f(x)/g(x)`
  • for derivatives
    `d/dx ((sin(x)* x^2)/(1 + tan(cot(x))))`
  • Multiple ways to derive functions
    `d/dy` cot(x*y)
  • Implicit derivatives, too
    `d/dx (y(x)^2 - 5*sin(x))`
  • the nth derivative formula
    ` d^n/dx^n (sin(x)*exp(x)) `

  • Integrals

  • click the ∫ button to integrate above result
    `int(cos(x)*e^x+sin(x)*e^x)` dx
  • `int tan(x) dx`
  • semi integrate, semiint()
    `int sin(x) dx^(1/2)`
  • Multiple integrate `int int (x + y) dx dy`
  • Definite integration `int _1^3 (2*x + 1) dx`
  • Improper integral `int _0^(pi/2) tan(x) dx`
  • Infinite integral `int _0^oo 1/(x^2 + 1) dx`
  • Exact answers `int (2x+3)^7 dx`
  • numeric computation by click on the "~=" button
    n( `int _0^1` sin(cos(x)) dx )
  • infinite integrate integrate(exp(-x) as x->oo)
    integrate
  • `int` sin(x) dx
  • integrate(sin(x))

    semiintegrate

  • `d^(-0.5)/dx^(-0.5) sin(x)`
  • semiint(sin(x))

    Definite integration

  • `int_1^2` sin(x) dx
  • integrate( sin(x) as x from 1 to 2 )
  • integrate sin(x) as x from 0 to 1

  • Equation >>

    Algebra Equation
  • solve equation and inequalities,
    solve( x^2+3*x+2 )

  • Symbolic roots
    solve( x^2 + 4*x + a )

  • Complex roots
    solve( x^2 + 4*x + 181 )

  • numerical root
    nsolve( x^3 + 4*x + 181 )

  • solve equation to x.
    solve( x^2-5*x-6=0 to x )
  • by default, equation = 0 to default unknown x.
    solve( x^2-5*x-6 )

  • system of 2 equations with 2 unknown x and y.
    solve( 2x+3y-1=0,x+y-1=0, x,y )

  • Diophantine equation
    number of equation is less than number of the unknown, e.g. one equation with 2 unknowns x and y.
    solve( 3x-10y-2=0, x,y )

  • Modulus equation
    mod(x-1,10)=2

  • congruence equation
    3x=2*(mod 10)
    3x-2=0 (mod 10)
    3x-2=(mod 10)

  • functional equation
    rsolve() solves recurrence equation to unknown y.
    y(x+1)-y(x)=x

  • Inequalities
    solve( 2*x-1>0 )
    solve( x^2+3*x+2>0 )

  • differential equation
    dsolve() solves differential equation to unknown y.
    y'=x*y+x
    y'= 2y
    y'-y-1=0
    (y')^2-2y^2-4y-2=0
  • dsolve also solves fractional differential equation
    `d^0.5/dx^0.5 y = 2y`
    `d^0.5/dx^0.5 y - exp(x-1)/(x+1)*y = 0`
  • dsolve( y' = sin(x-y) )
  • dsolve( y(1,x)=cos(x-y) )
  • dsolve( ds(y)=tan(x-y) )

  • integral equation
    `int y \ dx = 2y`
    `int_0^x y(t)/sqrt(x-t)` dt = 2y`

  • fractional differential equation
    dsolve( `(d^0.5y)/dx^0.5=sin^((-0.5))(x)` )

  • fractional integral equation
    `d^-0.5/dx^-0.5 y = 2y`

  • test solution for differential equation by odetest() or test().
    odetest( exp(2x), `dy/dx=2y` )
    odetest( exp(4x), `(d^0.5y)/dx^0.5=2y` )

  • Series >>

  • convert to sum series definition
    tosum( sin(x) )
  • check its result by simplify()
    simplify( tosum( sin(x) ))
  • expand above sum series
    expand( tosum(sin(x)) )
  • compare to Taylor series
    taylor( sin(x), x=0, 8)
  • compare to series
    series( sin(x) )

  • Taylor series expansion as x=0,
  • taylor( exp(x) as x=0 )
    by default x=0,
  • taylor(exp(x))

    series expand not only to taylor series,

  • series( exp(x) )
    but aslo to other series expansion,
  • series( zeta(2,x) )

  • Discrete Mathematics >>

    default index variable in discrete math is k.
  • Difference Δ`k^2`

  • Summation ∑

  • Indefinite sum ∑ k
  • Check its result by difference Δ`sum k`
  • Definite sum, Partial sum x from 1 to x, e.g. 1+2+ .. +x =
    `sum _(k=1) ^x k`
  • Definite sum, sum x from 1 to 5, e.g. 1+2+ .. +5 =
    ∑(x,x,0,5)
  • Infinite sum x from 0 to inf, e.g. 1/0!+1/1!+1/2!+ .. +1/x! =
    sum 1/(x!) as x->oo
  • sum(x^k,k,0,5)
  • sum(2^k, k,0, x)
  • cpnvert to sum series definition
    tosum( sin(x) )
  • expand above sum series
    expand( tosum(sin(x)) )

  • Indefinite sum
  • ∑ k
  • sum( x^k/k!,k )

    partial sum of 1+2+ .. + k = ∑ x

  • partialsum(k)

    Definite sum of 1+2+ .. +5 = ∑ x

  • sum(x,x,0,5,1)

    Infinite sum of 1/0!+x/1!+ .. +x^k/k! = sum( x^k/k! as k->oo )

  • infsum( x^k/k!,k )

    Product ∏

  • prod(x)

  • `prod x`

    Definition >>

  • definition of function
    definition( sin(x) )
  • check its result by simplify()
    simplify( def(sin(x)) )
  • convert to sum series definition
    tosum( sin(x) )
  • check its result by simplify()
    simplify( tosum(sin(x)) )
  • convert to integral definition
    toint( sin(x) )
  • check its result by simplify()
    simplify( toint(sin(x)) )
  • Number Theory >>

  • double factorial 6!!
  • Calculate the 4nd prime prime(4)
  • is prime number? isprime(12321)
  • next prime greater than 4 nextprime(4)
  • binomial number `((4),(2))`
  • combination number `C_2^4`
  • harmonic number `H_4`
  • congruence equation 3x=2*(mod 10)
  • 3x-2=0 (mod 10), 3x-2=(mod 10)
  • modular equation mod(x-1,10)=2
  • Diophantine equation
    number of equation is less than number of the unknown, e.g. one equation with 2 unkowns x and y,
    solve( 3x-10y-2=0, x,y )
  • Plot >>

  • plot sin(x) to show solution, by moving mouse wheel to zoom
    plot( sin(x) )
  • plot sin(x) and x^2 to show solutions on cross
    plot( sin(x) and x^2)
  • implicit plot sin(x)=y to show a multivalue function, by moving mouse wheel to zoom
    plot( sin(x)=y )
  • parametric plot with default pararmter t
    parametricplot( sin(t) and sin(4*t) )
  • polar plot
    polarplot( 2*sin(4*x) )
  • Geometry >>

  • tangent plot, by moving mouse on the curve to show tangent
    tangentplot( sin(x) )
  • secant plot, by moving mouse on the curve to show secant
    secantplot( sin(x) )
  • circle with radius 2
    circle(2)
  • plane curve
  • 


    See Also


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