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Examples of Fractional Calculus Computer Algebra System

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Arithmetic 算术 >>

Exact computation

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

  • Complex

  • input complex number in polar(r,theta*degree) coordinates
    polar(1,45degree)

  • input complex number in polar(r,theta) coordinates for degree by polard(r,degree)
    polard(1,45)

  • input complex number in r*cis(theta*degree) format
    2cis(45degree)
  • Convert to complex
    tocomplex(polar(1,45degree))
  • Convert complex a+b*i to polar(r,theta) coordinates
    convert 1-i to polar = topolar(1-i)
  • Convert complex a+b*i to polar(r,theta*degree) coordinates
    topolard(1-i)

  • Numerical approximations

  • Convert back by numeric computation n()
    n(polar(2,45degree))
    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 代数 >>

  • 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) )

  • Function 函数 >>

  • Function 函数
  • Trigonometry 三角函数
  • inverse
    inverse( sin(x) )

  • plot a multivalue function
    inverse( sin(x)=y )

  • expand
    expand( sin(x)^2 )
  • factor
    factor( sin(x)*cos(x) )
  • Calculus 微积分 >>

    Limit

    `lim_(x->0) sin(x)/x ` = lim sin(x)/x as x->0 = lim(sin(x)/x)
    `lim _(x->oo) log(x)/x` = lim( log(x)/x as x->inf )


    Derivatives

    Differentiate

    `d/dx sin(x)` = d(sin(x))

  • Second order derivative
    `d^2/dx^2 sin(x)` = d(sin(x),x,2) = d(sin(x) as x order 2)

  • sin(0.5,x) is inert holder of the 0.5 order derivative `sin^((0.5))(x)`, it can be activated by evaulate() or simplify():
    evaulate( sin(0.5,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 order 2) = d(x^6, x=1, 2)

    Fractional calculus

  • semiderivative
    `d^(0.5)/dx^(0.5) sin(x)` = d(sin(x),x,0.5) = d( sin(x) as x order 0.5) = semid(sin(x))

  • input sin(0.5,x) as the 0.5 order derivative of sin(x) for
    `sin^((0.5))(x)` = `sin^((0.5))(x)` = sin(0.5,x)
  • simplify sin(0.5,x) as the 0.5 order derivative of sin(x),
    `sin^((0.5))(x)` = simplify(sin(0.5,x))
  • 0.5 order derivative again
    `d^(0.5)/dx^(0.5) d^(0.5)/dx^(0.5) sin(x)` = d(d(sin(x),x,0.5),x,0.5)
  • Minus order derivative
    `d^(-0.5)/dx^(-0.5) sin(x)` = d(sin(x),x,-0.5)

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

  • click the ∫ button to integrate above result
    `int(cos(x)*e^x+sin(x)*e^x)\ dx` = int(cos(x)*e^x+sin(x)*e^x)
    `int tan(x)\ dx` = integrate tan(x) = int(tan(x))
  • semi integrate, semiint()
    `int sin(x) \ dx^(1/2)` = int(sin(x),x,1/2) = int sin(x) as x order 1/2 = semiint(sin(x)) = d(sin(x),x,-1/2)
  • Multiple integrate
    `int int (x + y)\ dx dy` = int( int(x+y, x),y)
    `int int exp(-x)\ dx dx` = integrate(exp(-x) as x order 2)
  • Definite integration
    `int _1^3` (2*x + 1) dx = int(2x+1,x,1,3) = int(2x+1 as x from 1 to 3)
  • Improper integral
    `int _0^(pi/2)` tan(x) dx =int(tan(x),x,0,pi/2)
  • Infinite integral
    `int _0^oo 1/(x^2 + 1)` dx = int(1/x^2+1),x,0,1)
  • Exact answers
    `int (2x+3)^7` dx = int (2x+3)^7
  • numeric computation by click on the "~=" button
    n( `int _0^1` sin(x) dx ) = nint(sin(x),x,0,1) = nint(sin(x))
  • infinite integrate

    integrate

  • `int` sin(x) dx = integrate(sin(x))

  • semiintegrate
    `int sin(x)\ dx^0.5` = `d^(-0.5)/dx^(-0.5) sin(x)` = int(sin(x),x,0.5) = semiint(sin(x))

  • Definite integration
    `int_0^1` sin(x) dx = integrate( sin(x),x,0,1 ) = integrate sin(x) as x from 0 to 1

  • Equation 方程 >>

    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 unknowns 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=2*(mod 10)
    3x-2=2mod(10)

  • functional equation
    rsolve() solves recurrence equation to unknown y.
    f(x+1)-f(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( 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
    `int_0^sqrt(x) (y(t^2))` dt = sin(x)

  • differential and integral equation
    ds(y)-ints(y) -y-exp(x)=0
    `dy/dx-int y dx -y-exp(x)=0`

  • fractional differential equation
    dsolve also solves fractional differential equation
    `d^0.5/dx^0.5 y = 2y`
    `d^0.5/dx^0.5 y -y - E_(0.5) (4x^0.5) = 0`
    `d^0.5/dx^0.5 y -y -exp(4x) = 0`
    `(d^0.5y)/dx^0.5=sin(x)`

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

  • fractional differential and integral equation
    ds(y,x,0.5)-ints(y,x,0.5) -y-exp(x)=0
    `(d^0.5y)/(dx^0.5)-int y (dx)^0.5 -y-exp(x)=0`

  • function order differential equation
    `(d^sin(x) y)/dx^sin(x)-y-exp(x)=0`
    `(d^cos(x) y)/dx^cos(x)-y-exp(x)=0`

  • system of 2 equations with 2 unknowns x of the 0.5 order and y of the 0.8 order with a variable t.
    dsolve( x(0.5,t)=t,y(0.8,t)=x )

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

  • 2000 examples of Ordinary differential equation (ODE)
  • Series 级数 >>

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

    by default x=0.

  • series expand not only to taylor series,
    series( exp(x) )
    but aslo to other series expansion,
    series( zeta(2,x) )

  • Discrete Math 离散数学 >>

    default index variable in discrete math is k.
  • Difference
    Δ`k^2` = difference(k^2)

    Summation ∑

  • Indefinite sum
    ∑ k = sum(k)
  • Check its result by difference
    Δ`sum k` = difference( sum(k) )
  • Definite sum, Partial sum x from 1 to x, e.g.
    1+2+ .. +x = `sum _(k=1) ^x k` = sum(k,k,1,x)
  • Definite sum, sum x from 1 to 5, e.g.
    1+2+ .. +5 = ∑(x,x,0,5) = sum(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( exp(x) )
  • expand above sum series
    expand( tosum(exp(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 定义式 >>

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

  • 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-1=2*(mod 10)
    3x-1=2mod( 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 )
  • Probability 概率 >>

  • P() is probability of standard normal distribution
    P(x<0.8)
  • Phi() is standard normal distribution function
    `Phi(x)`
  • Plot 制图 >>

    Plot 制图
  • plot sin(x) to show solution, by moving mouse wheel to zoom
    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
    implicitplot( x=sin(y) )
  • parametric plot with default pararmter t
    parametricplot( sin(t) and sin(4*t) )
  • polar plot
    polarplot( 2*sin(4*x) )

  • Interactive 互动
  • 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) )
  • Geometry 几何 >>

  • semicircle with radius 2, 半园
    semicircle(2)
  • circle with radius 2, 园
    circle(2)
  • oval with x radius 2 and y radius 1, 椭园
    oval(2,1)

  • Interactive 互动
  • tangent 切线 as x=1
    tangent( sin(x) as x=1 )
    切线 by default, at x=0
    tangent( sin(x) )
  • secantnt 割线 at x=0
    secantplot( sin(x) )
  • 3D graph 立体图 plot3D

    Animation 动画

    programming 编程 >>

    online programming 在线编程
    1. plot 函数图
    2. rose 玫瑰花
    3. check code 验证码
    4. calculator 计算器
    5. sci calculator 科学计算器
    6. color 颜色取色器

      Animation 动画
    
    See Also

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