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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) )

  • Trigonometry 三角函数 >>

    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 activate() or simplify():
    activate( 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 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
    `(d^0.5y)/dx^0.5=sin(x)`

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

  • 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<1)
  • 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) )
  • Animation 动画

    3D graph 立体图 plot3d

    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
    7. solar 太阳系行星运行
    8. solar system 太阳系行星运行
    9. taiji 太极图
    10. taichi 八卦太极图
    11. clock 时钟
    12. html clock html 时钟
    13. ccs clock ccs 时钟
    14. heart beat 心跳
    15. love u 我爱你
    16. percent 进度百分比
    17. new year 新年
    18. driving dog 会开车的狗
    19. sea
    20. fish jumping 鱼跳水

      Interactive 互动
    21. solar 3D 太阳系行星运行
    22. water 可以升降的水
    23. sea water 可以升降的海水
    24. solar eclipse 日食
    25. broke net 打破网
    26. love tree 爱心树
    
    See Also

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