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Examples of Fractional Calculus Computer Algebra System


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Arithmetic >>

Exact computation

  • Fraction `1-1/2`
  • Big number 100!

  • 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)-polar(4,45degree)

  • Convert to complex
    tocomplex(polar(3,45degree)-polar(4,45degree))

  • 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 )
  • Trigonometry >>

    Calculus >>

    Limits

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

  • Derivatives

    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)`
  • 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 )
  • Series >>

  • convert to sum series definition
    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) )
  • Number Theory >>

  • factorial 106!
  • Calculate the 4nd prime prime(4)
  • is prime number? isprime(12321)
  • First prime greater than 4 nextprime(4)
  • binomial number `C_2^4`
  • harmonic number `H_4`
  • Discrete Mathematics >>

  • Difference Δ`x^2`

  • Summation ∑

  • Indefinite sum ∑x
  • 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
  • Infinite sum x from 0 to inf, e.g. 1/0!+1/1!+1/2!+ .. +1/x! =
    `sum_0^oo 1/(x!)`
  • 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)) )

  • Product ∏

    prod(x)
  • `prod x`
  • Miscellaneous >>

  • definition of function
    definition( sin(x) )
  • simplify definition of function
    simplify( def(sin(x)) )
  • conver to sum series definition
    tosum( sin(x) )
  • simplify series definition of function
    simplify( tosum(sin(x)) )
  • inverse the 0.5 order derivative of sin(x) function
    (-1)`( sin^0.5(x) )`
  • Plot >>

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



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