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Examples of Fractional Calculus Computer Algebra System 例题

Content

  • Arithmetic 算术
    1. Exact_computation
    2. Numerical approximations
    3. Complex 复数
  • Numeric math 数值数学
  • Algebra 代数
    1. expand
    2. factor
    3. combine
    4. convert
    5. inverse function
    6. inverse equation
    7. polynomial
  • Function 函数
    1. Trigonometry 三角函数
    2. Complex Function 复变函数
    3. special Function
  • Calculus 微积分
    1. Limit
    2. Derivatives
    3. Integrals
    4. Fractional calculus
    5. vector calculus
  • Equation 方程
    1. inverse equation
    2. polynomial equation
    3. Algebra_equation
    4. absolute equation
    5. Diophantine equation
    6. congruence equation
    7. Modulus equation
    8. Probability_equation
    9. recurrence_equation
    10. functional_equation
    11. difference_equation
    12. vector_equation
    13. system of equations
    14. 2D equations
    15. 2D parametric equations
    16. 3D equations
    17. 3D parametric equations
    18. 4D equations

    19. Inequalities

    20. differential equation
    21. ordinary differential equation (ODE)
    22. fractional differential equation
    23. integral equation
    24. fractional integral equation
    25. differential integral equation
    26. system of differential equations

    27. partial differental equation (PDE)
    28. fractional partial differental equation
    29. system of partial differential equations
    30. test solution
  • bugs
  • Transform 转换
    1. laplace
    2. fourer
    3. convolute
  • Discrete Math 离散数学
    1. Summation ∑
    2. Indefinite sum
    3. definite sum
    4. infinite sum
    5. Series 级数
    6. Product ∏
  • Definition 定义式
  • Number Theory 数论
  • Probability 概率
  • Multi elements 多元
    1. and
    2. list
  • Statistics 统计
  • Linear Algebra 线性代数
    1. vector
    2. matrix
    3. array
  • programming 编程
  • Graphics
    1. Classification by plotting function 按制图函数分类
    2. Classification by dimension 按维数分类
    3. Classification by appliaction 按应用分类
    4. Classification by objects 按物体分类
    5. Classification by function 按函数分类
    6. Classification by equation 按方程分类
    7. Classification by domain 按领域分类
    8. Classification by libary 按文库分类
    9. Classification by platform 按平台分类
  • Plot 制图
    1. Interactive plot 互动制图
    2. parametric plot, polar plot
    3. solve equation graphically
    4. area plot with integral
    5. complex plot
    6. Geometry 几何
  • plane graph 平面图
    1. plane graph 平面图 with plot2D
    2. function plot with funplot
    3. differentiate graphically with diff2D
    4. integrate graphically with integrate2D
    5. solve ODE graphically with odeplot
  • 3D graph 立体图
    1. surface in 3D with plot3D
    2. contour in 3D with contour3D
    3. wireframe in 3D with wirefram3D
    4. complex function in 3D with complex3D
    5. a line in 3D with parametric3D
    6. a column in 3D with parametric3D
    7. the 4-dimensional object (x,y,z,t) in 3D with implicit3D
  • Drawing 画画

    Do exercise and learn from example.

      Arithmetic 算术 >>

      Exact computation

    1. Fraction `1E2-1/2`
    2. mod operation:
      input mod(3,2) for 3 mod 2

      big number in Java

    3. Add prefix "big" to number for Big integer:
      big1234567890123456789-1
    4. Add prefix "big" to number for Big decimal:
      big1.234567890123456789-1
    5. big number in JavaScript

      computation in JavaScript should to click the numeric button or expression ending with equal sign =

      The default precision for BigNumber is 64 digits, and can be configured with the option precision.

    6. bignumber(1.23456789123456789)=

      Round-off errors

      Calculations with BigNumber are much slower than calculations with Number, but they can be executed with an arbitrary precision. By using a higher precision, it is less likely that round-off errors occur:
      // round-off errors with numbers
    7. math.add(0.1, 0.2) // Number, 0.30000000000000004
    8. math.divide(0.3, 0.2) // Number, 1.4999999999999998

      // no round-off errors with BigNumbers :)

    9. math.add(math.bignumber(0.1), math.bignumber(0.2)) // BigNumber, 0.3
    10. math.divide(math.bignumber(0.3), math.bignumber(0.2)) // BigNumber, 1.5

      Numerical approximations

      There are two types of numeric computation: JavaScript and Java numeric computation:

      JavaScript numeric computation

    11. numeric computation with the ≈ button :
      acos(bignumber(-1))

    12. numeric computation end with the equal sign = :
      acos(bignumber(-1))=

      Java numeric computation

    13. numeric computation with the ≈≈ button :
      sin(pi/4)

    14. Convert back with numeric computation function n(x) :
      n(polar(2,45degree))
      n( sin(pi/4) )
      n( sin(30 degree) )
    15. `sin^((0.5))(1)` is the 0.5 order derivative of sin(x) at x=1 :
      n( sin(0.5,1) )
    16. `sin(1)^(0.5)` is the 0.5 power of sin(x) at x=1 :
      n( sin(1)^0.5 )
    17. more are in numeric math

      Complex 复数

      math handbook chapter 1.1.2 complex
      There are two types of complex number: JavaScript complex and Java complex:

      input complex numbers

    18. input complex numbers in the complex plane:
      1+2i
    19. in small letters of complex(1,2) is complex number:
      complex(1,2)
    20. in first upper case letter of Complex(1,2) is object in JavaScript complex:
      Complex(1,2)

    21. input complex number in polar(r,theta) coordinates:
      polar(1,pi)

    22. input complex number in polar(r,theta*degree) coordinates:
      polar(1,45degree)

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

    24. input complex number in trig format of r*cis(theta*degree) :
      2cis(45degree)
    25. Convert complex to polar(r,theta) coordinates

    26. Convert complex a+b*i to polar(r,theta) coordinates in Java:
      convert 1-i to polar
      topolar(1-i)
      convert back to a+b*i format by to remove last letter s or click the simplify button

    27. Convert complex a+b*i to polar(r,theta*degree) coordinates:
      topolard(1-i)

    28. convert back to a+b*i format by to remove last letter s or click the simplify button

    29. Convert complex a+b*i to polar(r,theta) coordinates in JavaScript:
      math.Complex(1,2).toPolar()=
      math.complex(1,2).toPolar()=

      convert to vector

      complex(1,2) number is special vector, i.e. the 2-dimentional vector, so it can be converted to vector.

    30. to Java vector(1,2)
      convert 1-i to vector
      tovector(1-i)

    31. to JavaScript vector [1,2]
      math.Complex(1,2).toVector()=
      math.complex(1,2).toVector()=

      complex number plot

    32. in order to auto plot complex number as vector, input complex(1,-2) for 1-2i, or convert 1-2i to complex(1,-2) by
      convert 1-2i to complex
      convert(1-2i to complex)
      tocomplex(1-2i)

      complex function plot

    33. complex animation(z) show animation of complex function in complex domain of complex variable z.
    34. complexplot(z) show phrase and/or modulus of complex function in complex domain of complex variable z.

    35. complex 2D plot:
      complex2D(x^x)

      more are in complex2D(x) show 2 curves of real and imag parts in real domain of real variable x..

    36. complex 3D plot:
      complex3D(pow(x,x))

      more are in complex function with complex3D(x) in complex domain of complex variable x.

    37. Numeric math 数值数学 >>

      数值数学 = 计算数学
      math handbook chapter 3.4

    38. computation in JavaScript should to click the numeric button or expression ending with equal sign =
      sin(pi/4)=

    39. numeric computation with the n(x) ≈≈ button:
      n( sin(30 degree) )
      n sin(30 degree)

    40. numeric solve equation:
      nsolve( x^2-5*x+6=0 )
      nsolve( x^2-5*x+6 )

    41. find_root(x,-10,10) between -10 and 10 for numeric equation:
      find_root( x^2-5*x+6==0 )
      find_root( x^2-5*x+6 )

    42. JavaScript numeric calculator with the ≈ button can calculate numeric, number theory, Probability, Statistics, matrix, solve equation.

      more example in JavaScript mathjs

    43. numeric integrate, by default x from 0 to 1:
      nint( x^2-5*x+6,x,0,1 )
      nint x^2-5*x+6 as x from 0 to 1
      nint sin(x)

    44. numeric computation with the funplot ≈ button:
      integrate(x=>sin(x),[1,2])

      more calculus operation in JavaScript calculus

    45. Algebra 代数 >>

      math handbook chapter 1 algebra

    46. simplify:
      taylor( (x^2 - 1)/(x-1) )
    47. expand:
      expand( (x-1)^3 )

    48. factorizing:
      factor( x^2+3*x+2 )
    49. factor high order polymonial by factor(x)== :
      factor( x^3-1 )==
    50. factorization:
      factor( x^4-1 )==
    51. combine two terms:
      combine( log(a)+log(b) )
    52. tangent

    53. tangent equation at x=0 by default
      tangent( sin(x) )

    54. tangent equation at x=1
      tangent( sin(x),x=1 )

    55. tangentplot(x) show dynamic tangent line when your mouse over the curve.
      tangentplot( sin(x) )

      convert

      convert( sin(x) to exp) is the same as toexp(sin(x))
    56. convert to exp:
      toexp( cos(x) )
    57. convert to trig:
      convert exp(x) to trig
    58. convert sin(x) to exp(x):
      convert sin(x) to exp = toexp( sin(x) )

    59. Convert to exp(x):
      toexp(Gamma(2,x))
    60. inverse function

    61. input sin(x), click the inverse button
      inverse( sin(x) )
      check its result by clicking the inverse button again.
      In order to show multi-value, use the inverse equation instead function.

      inverse equation

    62. inverse equation to show multivalue if it has:
      inverse( sin(x)=y )
      check its result by clicking the inverse button again.

      polynomial

      math handbook chapter 20.5 polynomial

    63. the unit polynomial:
      Enter poly(3,x) = poly(3) for the unit polynomial with degree 3: x^3+x^2+x+1.

    64. Hermite polynomial:
      hermite(3,x) gives the Hermite polynomial while hermite(3) gives Hermite number.

    65. harmonic polynomial:
      harmonic(-3,1,x) = harmonic(-3,x)

    66. the zeta( ) polynomial:
      zeta(-3,x)

    67. simplify(x):
      taylor( (x^2 - 1)/(x-1) )
    68. expand(x) polynomial:
      expand(hermite(3,x))

    69. topoly(x) convert polynomial to polys(x) as holder of polynomial coefficients,
      convert `x^2-5*x+6` to poly = topoly( `x^2-5*x+6` )

    70. simplify polys(x) to polynomial:
      simplify( polys(1,-5,6,x) )

    71. polyroots(x) is holder of polynomial roots, topolyroot(x) convert a polynomial to polyroots.
      convert (x^2-1) to polyroot = topolyroot(x^2-1)

    72. polysolve(x) numerically solve polynomial for multi-roots:
      polysolve(x^2-1)

    73. nsolve(x) numerically solve for a single root:
      nsolve(x^2-1)

    74. solve(x) for sybmbloic and numeric roots:
      solve(x^2-1)

    75. construct polynomial from roots, activate polyroots(x) to polynomial, and reduce roots to polynomial equation = 0 by click on the simplify(x) button.
      simplify( polyroots(2,3) )

      Number

      When the variable x of polynomial is numnber, it becomes polynomial number, please see Number_Theory section.
    76. Function 函数 >>

      math handbook chapter 1 Function 函数

      Trigonometry 三角函数

    77. expand Trigonometry by expandtrig(x) :
      expandtrig( sin(x)^2 )
    78. inverse function :
      inverse( sin(x) )

    79. plot a multivalue function by the inverse equation :
      inverse( sin(x)=y )

    80. expand trig function :
      expandtrig( sin(x)^2 )
    81. expand special function :
      expand( gamma(2,x) )
    82. factor(x) :
      factor( sin(x)*cos(x) )

    83. Complex Function 复变函数

      math handbook chapter 10 Complex Function 复变函数
      complex2D(x) shows 2 curves of the real and imag parts in real domain x, and complex3D(x) shows complex function in complex domain x, for 20 graphes in one plot.

    84. complex 2D plot :
      complex2D(x^x)

    85. put imagary i iwth variable x for auto complex 2D plot :
      exp(i*x)

      more are in complex2D(x) for 2 curves of real and imag parts in complex domain of complex variable x.

    86. complex animation(z) show animation of complex function in complex domain of complex variable z.
    87. complexplot(z) show phrase and/or modulus of complex function in complex domain of complex variable z.

    88. complex 3D plot :
      complex3D(pow(x,x)) in complex domain of complex variable x.

      Complex

      1. complex - complex function - complex math
      2. math handbook chapter 10 complex function
      3. complex animate(z) for phase animation, the independent variable must be z.
      4. complex plot(z) for phase and/or modulus, the independent variable must be z.
      5. complex2D(x) for complex 2 curves of real and imag parts, the independent variable must be x.
      6. complex3D(x) for 3 dimensional graph, the independent variable must be x.
      7. color WebXR surface of complex function on complex plane
      8. Riemann surface - Complex Branches - complex coloring

      References

      1. Complex analysis
      more are in complex function

      special Function

      math handbook chapter 12 special Function
    89. Calculus 微积分 >>

      Limit

      math handbook chapter 4.1 limit

    90. click the lim(x) button for Limit at x->0 :
      `lim_(x->0) sin(x)/x ` = lim sin(x)/x as x->0 = lim(sin(x)/x)

    91. click the nlim(x) button for numeric limit at x->0 :
      nlim(sin(x)/x)

    92. click the limit(x) button for Limit at x->oo :
      `lim _(x->oo) log(x)/x` = lim( log(x)/x as x->inf )
      = limoo( log(x)/x )

    93. one side limit, left or right side :
      lim(exp(-x),x,0,right)

      Derivatives

      Math Handbook chapter 5 differential calculus

    94. Differentiate
      `d/dx sin(x)` = d(sin(x))

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

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

    97. Derivative as x=1 :
      `d/dx | _(x->1) x^6` = d( x^6 as x->1 )

    98. 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

      Fractional calculus

    99. 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))

    100. 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)

    101. simplify sin(0.5,x) as the 0.5 order derivative of sin(x) :
      `sin^((0.5))(x)` = simplify(sin(0.5,x))

    102. 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)

    103. Minus order derivative :
      `d^(-0.5)/dx^(-0.5) sin(x)` = d(sin(x),x,-0.5)

    104. inverse the 0.5 order derivative of sin(x) function :
      f(-1)( sin(0.5)(x) ) = inverse(sin(0.5,x))
    105. Derive the product rule :
      `d/dx (f(x)*g(x)*h(x))` = d(f(x)*g(x)*h(x))
    106. … as well as the quotient rule :
      `d/dx f(x)/g(x)` = d(f(x)/g(x))
    107. for derivatives :
      `d/dx ((sin(x)* x^2)/(1 + tan(cot(x))))` = d((sin(x)* x^2)/(1 + tan(cot(x))))
    108. Multiple ways to derive functions :
      `d/dy cot(x*y)` = d(cot(x*y) ,y)
    109. Implicit derivatives, too :
      `d/dx (y(x)^2 - 5*sin(x))` = d(y(x)^2 - 5*sin(x))
    110. the nth derivative formula :
      ` d^n/dx^n (sin(x)*exp(x)) ` = nthd(sin(x)*exp(x))
    111. differentiate graphically

      some functions cannot be differentiated or integrated symbolically, but can be semi-differentiated and integrated graphically in diff2D.

      Integrals

      Math Handbook chapter 6 integral calculus
    112. indefinite integrate : `int` sin(x) dx = integrate(sin(x))

    113. enter a function sin(x), then click the ∫ button to integrate :
      `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))

    114. Exact answers for integral :
      `int (2x+3)^7` dx = int (2x+3)^7

    115. 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)

    116. Definite integration :
      `int _1^3` (2*x + 1) dx = int(2x+1,x,1,3) = int(2x+1 as x from 1 to 3)

    117. Improper integral :
      `int _0^(pi/2)` tan(x) dx =int(tan(x),x,0,pi/2)

    118. Infinite integral :
      `int _0^oo 1/(x^2 + 1)` dx = int(1/x^2+1),x,0,oo)

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

      integrator

      If integrate(x) cannot do, please try integrator(x) :
    120. integrator(sin(x))

    121. enter sin(x), then click the ∫ dx button to integrator

      fractional integrate

    122. semi integrate, semiint(x) :
      `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)

    123. indefinite semiintegrate :
      `int sin(x)\ dx^0.5` = `d^(-0.5)/dx^(-0.5) sin(x)` = int(sin(x),x,0.5) = semiint(sin(x))

    124. Definite fractional integration :
      `int_0^1` sin(x) `(dx)^0.5` = integrate( sin(x),x,0.5,0,1 ) = semiintegrate sin(x) as x from 0 to 1

      numeric computation

    125. numeric computation by click on the "~=" button :
      n( `int _0^1` sin(x) dx )

      numeric integrate

      If numeric computation ail, please try numeric integrate nintegrate(x) or nint(x) :
      nint(sin(x),x,0,1) = nint(sin(x))

      integrate graphically

      some functions cannot be differentiated or integrated symbolically, but can be semi-differentiated and integrated graphically in integrate2D.

      vector calculus

    126. differentiate vector(x,x) :
      d(vector(x,x))

    127. differentiate sin(vector(x,x)) :
      d(sin(vector(x,x)))
    128. Equation 方程 >>

      equation world

      Your operation step by step

    129. solve x^2-1=0 step by step
      1. add ( ) to equation, then add +1:
        (x^2-1=0)+1
      2. add ( ) to equation, then power by 0.5:
        (x^2=1)^0.5

      inverse an equation

    130. inverse an equation to show multivalue curve.
      inverse( sin(x)=y )
      check its result by clicking the inverse button again.

      polynomial equation

    131. polyroots(x) is holder of polynomial roots, topolyroot(x) convert a polynomial to polyroots.
      convert (x^2-1) to polyroot = topolyroot(x^2-1)

    132. polysolve(x) numerically solve polynomial for multi-roots.
      polysolve(x^2-1)

    133. construct polynomial from roots, activate polyroots(x) to polynomial, and reduce roots to polynomial equation = 0 by click on the simplify(x) button.
      simplify( polyroots(2,3) )

    134. solve(x) for sybmbloic and numeric roots :
      solve(x^2-1)
      solve( x^2-5*x-6 )

    135. solve equation and inequalities, by default, equation = 0 for default unknown x if the unknown omit.
      solve( x^2+3*x+2 )
    136. Symbolic roots :
      solve( x^2 + 4*x + a )

    137. Complex roots :
      solve( x^2 + 4*x + 181 )

    138. solve equation for x.
      solve( x^2-5*x-6=0,x )

    139. numerically root :
      nsolve( x^3 + 4*x + 181 )

    140. nsolve(x) numerically solve for a single root.
      nsolve(x^2-1)

    141. find_root(x) numerically find for a single root.
      find_root(x^2-1)

      Algebra Equation f(x)=0

      math handbook chapter 3 algebaic Equation

      solve(x) also solve other algebra equation, e.g.

    142. nonlinear equations:
      solve(exp(x)+exp(-x)=4)
      solve cos(x)+sin(x)=1

      if solve(x) cannot solve, then click the numeric button to solve numerically.

    143. solve(cos(x)-sin(x)=1)
      then click the numeric button

      absolute equation

    144. solve(x) absolute equation for the unknown x inside the abs(x) function :
      input abs(x-1)+abs(x-2)=3 for
      |x-1|+|x-2|=3
      click the solve button

      Modulus equation

      solve(x) Modulus equation for the unknown x inside the mod(x) function :
      first order equation
    145. input mod(3x,5)=1 for
      3x mod 5 = 1
      click the solve button. it is solved by inversemod(3,5)

      second order equation
    146. Enter mod(x^2-5x+7,2)=1 for
      (x^2-5x+7) mod 2 = 1

    147. Enter mod(x^2-5x+6,2)=0 for
      (x^2-5x+6) mod 2 = 0

      congruence equation

      a x ≡ b* (mod m)

      math handbook chapter 20.3 congruence

      a x ≡ b* (mod m) means two remindars in both sides of equation are the same, i.e. congruence, it is the same as the modular equation mod(a*x,m)=mod(b,m). if b=1, the modular equation mod(a*x,m)=1 can be solved by inversemod(a*x,m). By definition of congruence, a x ≡ b* (mod m) if a x − b is divisible by m. Hence, a x ≡ b (mod m) if a x − b = m y, for some integer y. Rearranging the equation to the equivalent form of Diophantine equation a x − m y = b, which can be solved by solve(a*x-m*y=b,x,y).

      first order equation:
    148. 3x = 1*(mod 5)
      mod(3x,5) = mod(1,5)
      second order equation:
    149. x^2+3x+2 = 1*(mod 11)
      x^2+3x+2 = 1 mod(11)
      x^2+3x+2 = mod(11)

      2D equations f(x,y) = 0

      Diophantine equation f(x,y) = 0

      math handbook chapter 20.5 polynomial

      It is that number of equation is less than number of the unknown, e.g. one equation with 2 unknowns x and y.

    150. One 2D equation f(x,y) = 0 for 2 integer solutions x and y
      solve( 3x-2y-2=0, x,y )
      solve( x^2-3x-2y-2=0, x,y )

      2D equation f(x,y) = 0 solved graphically

      One 2D equation for 2 unknowns x and y, f(x,y) = 0, solved graphically by implicitplot(x)
    151. solve x^2-y^2=1 graphically
      x^2-y^2-1=0

      2D parametric equations x=f(t), y=g(t)

      A system of 2 equations with a parameter t for 2 unknowns x and y, x=f(t), y=g(t), solved graphically :
    152. parametricplot( x=cos(t), y=sin(t) )
    153. parametric3D( cos(t),sin(t) )
    154. parametric2D( cos(t),sin(t) )

      2D parametric equations x=f(u,v), y=f(u,v)

      A system of 3 equations with 2 parameters u and v for 3 unknowns x and y and z, x=f(u,v), y=f(u,v) solved graphically :
    155. parametric3D( cos(u*v),sin(u*v),u*v )
    156. wireframe3D( cos(x*y),sin(x*y) )
    157. parametric3D( cos(t),sin(t) )
    158. 2D surface

      3D equations

      3D parametric equations x=f(t), y=f(t), z=f(t)

      A system of 3 equations with a parameter t for 3 unknowns x and y and z, x=f(t), y=f(t), z=f(t), solved graphically :
    159. parametric3D( t,cos(t),sin(t) )

      3D parametric equations x=f(u,v), y=f(u,v), z=f(u,v)

      A system of 3 equations with 2 parameters u and v for 3 unknowns x and y and z, x=f(u,v), y=f(u,v), z=f(u,v), solved graphically :
    160. parametric3D( u,u-v,u*v )
    161. parametric surface

      One 3D equation f(x,y,z) = 0

      One equation for 3 unknowns x and y and z, f(x,y,z) = 0, solved graphically :
    162. implicit3D( x-y-z )
    163. plot3D( x-y-z )

      4D equations

      One 4D equation with 4 variables,
    164. f(x,y,z,t) = 0, solved graphically :
      implicit3D( x-y-z-t )

    165. f(x,y,n,t) = 0, solved graphically :
      plot2D( x-y-n-t )

      Probability_equation

    166. solve(x) Probability equation for the unknown k inside the Probability function P(x),
      solve( P(x>k)=0.2, k)

      recurrence_equation

    167. rsolve(x) recurrence and functional and difference equation for y(x)
      y(x+1)+y(x)+x=0
      y(x+1)+y(x)+1/x=0

    168. fsolve(x) recurrence and functional and difference equation for f(x)
      f(x+1)=f(x)+x
      f(x+1)=f(x)+1

      functional_equation

    169. rsolve(x) recurrence and functional and difference equation for y(x)
      y(a+b)=y(a)*y(b)
      y(a*b)=y(a)+y(b)

    170. fsolve(x) recurrence and functional and difference equation for f(x)
      f(a+b)=f(a)*f(b)
      f(a*b)=f(a)+f(b)

      difference equation

    171. rsolve(x) recurrence and functional and difference equation for y(x)
      y(x+1)-y(x)=x
      y(x+2)-y(x+1)-y(x)=0

    172. fsolve(x) recurrence and functional and difference equation for f(x)
      f(x+1)-f(x)=x
      f(x+2)- f(x+1)-f(x)=0

      vector equation

      see vector

      Inequalities

    173. solve(x) Inequalities for x.
      solve( 2*x-1>0 )
      solve( x^2+3*x+2>0 )

      system of equations f(x,y)=0, g(x,y)=0

      math handbook chanpter 4.3 system of equations

    174. system of 2 equations f(x,y)=0, g(x,y)=0 for 2 unknowns x and y by default if the unknowns omit with the solve() button :
      solve( 2x+3y-1=0, 3x+2y-1=0 )

    175. system of 2 equations f(x,y)=0 and g(x,y)=0 for 2 unknowns x and y by default if the unknowns omit. On First graph it is solved graphically, where their cross is solution:
      ( 2x+3y-1=0 and 3x+2y-1=0 )

      differential equation

      Math handbook chapter 13 differential equation.
      There are two types of differential equations: a single variable is ordinary differential equation (ODE) and multi-variables is partial differential equation (PDE).

      ordinary differential equation (ODE)

      ODE(x) and dsolve(x) and lasove(x) solve ordinary differential equation (ODE) for unknown y.

      Your operation step by step

    176. solve dy/dx=x/y step by step
      1. add ( ) to equation, then time by y:
        (dy/dx=x/y)*y
      2. add ( ) to equation, then time by dx:
        (dy/dx*y=x)*dx
      3. integrate by click the integrate button
        int(dy*y=dx*x)
      4. add ( ) to equation, then time by 2:
        (1/2y^2=1/2x^2)*2
      5. add ( ) to equation, then power by 0.5:
        (y^2=x^2)^0.5
    177. linear ordinary differential equations:
      dsolve y'=x*y+x
      ode y'= 2y
      ode y'-y-1=0

    178. nonlinear ordinary differential equations:
      ode (y')^2-2y^2-4y-2=0
      dsolve( y' = sin(x-y) )
      dsolve( y(1,x)=acos(y)-sin(x)-x )
      dsolve( ds(y)-cos(x)=asin(y)-x )
      dsolve( ds(y)=exp(y)-exp(x) )
      dsolve( ds(y)-exp(x)=log(y)-x )
      ode `y'-exp(y)-1/x-x=0`
      ode `y'-sinh(y)+x-1/sqrt(1+x^2)=0`
      ode `y'-tan(y)+x-1/(1+x^2)=0`

    179. second order nonlinear ordinary differential equations:
      dsolve( ds(y,x,2)=asin(y)-sin(x)-x )
      dsolve( ds(y,x,2)-exp(x)=log(y)-x )

    180. 2000 examples of Ordinary differential equation (ODE)

      more examples in bugs

      solve graphically

      The odeplot(x) can be used to visualize individual functions, First and Second order Ordinary Differential Equation over the indicated domain. Input the right hand side of Ordinary Differential Equations, y"=f(x,y,z), where z for y', then click the checkbox. by default it is first order ODE.
    181. second order ODE
      odeplot y''=y'-y

      integral equation

      Math handbook chapter 15 integral equation.

      indefinite integral equation

      indefinite integral equation
    182. linear equation
      input ints(y) -2y = exp(x) for
      ode `int y` dx - 2y = exp(x)

    183. nonlinear equation
      input ints(y) -2y^2 = 0 for
      ode `int y` dx - 2y^2 = 0

      double integral equation

      input ints(y,x,2) for double integral
    184. linear equation
      ode( `int int y` dx -y = exp(x) )

    185. nonlinear equation
      ode( `int int y` dx *y= exp(x) )

      definite integral equation

      definite integral equation
    186. linear equation
      input integrates(y(t)/sqrt(x-t),t,0,x) = 2y for
      ode `int_0^x (y(t))/sqrt(x-t)` dt = 2y

    187. nonlinear equation
      input integrates(y(t)/sqrt(x-t),t,0,x) = 2y^2 for
      ode `int_0^x (y(t))/sqrt(x-t)` dt = 2y^2

      differential integral equation

    188. input ds(y)-ints(y) -y-exp(x)=0 for
      ode `dy/dx-int y dx -y-exp(x)=0`

      fractional differential equation

      dsolve(x) also solves fractional differential equation
    189. linear equations:
      ode `d^0.5/dx^0.5 y = 2y`
      ode `d^0.5/dx^0.5 y -y -exp(4x) = 0`
      ode `(d^0.5y)/dx^0.5 -y=x`
      ode `d^0.5/dx^0.5 y -y - E_(0.5) (x^0.5) *x = 0`

    190. nonlinear equations:
      ode `(d^0.5y)/dx^0.5 = y^2*exp(x)`
      ode `(d^0.5y)/dx^0.5 = sin(y)*exp(x)`
      ode `(d^0.5y)/dx^0.5 = exp(y)*exp(x)`
      ode `(d^0.5y)/dx^0.5 = log(y)*exp(x)`
      ode `(d^0.5y)/dx^0.5 - a*y^2-b*y-c = 0`
      ode `(d^0.5y)/dx^0.5 - log(y) - exp(x) + x=0`
      ode `(d^(1/2)y)/dx^(1/2)-asin(y)+x-sin(x+pi/4)=0`

      linear fractional integral equation

    191. ode `d^-0.5/dx^-0.5 y = 2y`

      nonlinear fractional integral equation

    192. ode `d^-0.5/dx^-0.5 y = 2y^2`

      fractional differential integral equation

    193. ds(y,x,0.5)-ints(y,x,0.5) -y-exp(x)=0
      ode `(d^0.5y)/(dx^0.5)-int y (dx)^0.5 -y-exp(x)=0`

      complex order differential equation

    194. ode `(d^(1-i) y)/dx^(1-i)-2y-exp(x)=0`

      variable order differential equation

    195. ode `(d^sin(x) y)/dx^sin(x)-2y-exp(x)=0`

      system of differential equations

      system of 2 equations with 2 unknowns x and y with a variable t :
    196. linear equations:
      dsolve( ds(x,t)=x-2y,ds(y,t)=2x-y )

    197. nonlinear equations:
      dsolve( ds(x,t)=x-2y^2,ds(y,t)=2x^2-y )

    198. the second order system of 2 equations with 2 unknowns x and y with a variable t :
      dsolve( x(2,t)=x,y(2,t)=2x-y )

    199. the 0.5th order system of 2 equations with 2 unknowns x and y with a variable t :
      dsolve( x(0.5,t)=x,y(0.5,t)=x-y )

      partial differental equation

      Math handbook chapter 14 partial differential equation.
      PDE(x) and pdsolve(x) solve partial differental equation with two variables t and x for y, then click the plot2D button to plot solution, pull the t slider to change the t value. click the plot3D button for 3D graph.

    200. linear equation:
      pde `dy/dt = dy/dx-2y`

    201. nonlinear equation:
      pde `dy/dt = dy/dx*y^2`
      pde `dy/dt = 2dy/dx-y^2`
      pde `(d^2y)/(dt^2) -2* (d^2y)/(dx^2)-y^2-2x*y-x^2=0`

      partial differential integral equation

    202. ds(y,t)-ints(y,x)-y-exp(x)=0
      pde `(dy)/(dt)-int y (dx) -y-exp(x)=0`

      fractional partial differental equation

      PDE(x) and pdsolve(x) solve fractional partial differental equation.

    203. linear equations:
      pde `(d^0.5y)/dt^0.5 = dy/dx-2y`

    204. nonlinear equations:
      pde `(d^0.5y)/dt^0.5 = 2* (dy)/dx*y^2`
      pde `(d^0.5y)/dt^0.5 = 2* (d^0.5y)/dx^0.5-y^2`
      pde `(d^1.5y)/(dt^1.5) + (d^1.5y)/(dx^1.5)-2y^2-4x*y-2x^2 =0`

      More examples are in Analytical Solution of Fractional Differential Equations

      fractional partial differential integral equation

    205. ds(y,t)-ints(y,x,0.5)-exp(x)=0
      pde `(dy)/(dt)-int y (dx)^0.5 -exp(x)=0`
    206. ds(y,t)-ints(y,x,0.5)+2y-exp(x)=0
      pde `(dy)/(dt)-int y (dx)^0.5 +2y-exp(x)=0`
    207. ds(y,t)-ds(y,x)-ints(y,x,0.5)+3y-exp(x)=0
      pde `(dy)/(dt)-dy/dy-int y (dx)^0.5 +3y-exp(x)=0`

      system of partial differential equations

      system of 2 equations with two variables t and x for 2 unknowns y and z:
    208. linear equations:
      pde( ds(y,t)-ds(y,x)=2z-2y,ds(z,t)-ds(z,x)=4z-4y )

    209. the second order system of 2 equations :
      pde( ds(y,t)-ds(y,x,2)=2z-2y,ds(z,t)-ds(z,x,2)=4z-4y )

    210. the 0.5th order system of 2 equations :
      pde( ds(y,t)-ds(y,x,0.5)=2z-2y,ds(z,t)-ds(z,x,0.5)=4z-4y )

      test solution

      test solution for algebaic equation

      test solution for algebaic equation to the unknown x by test( solution,eq, x) or click the test button :
    211. test(1,x^2-1=0,x)
      test( -1, x^2-5*x-6 )

      test solution for differential equation

      test solution for differential equation to the unknown y by test( solution, eq ) or click the test button :
    212. test( exp(2x), `dy/dx=2y` )
    213. test( exp(4x), `(d^0.5y)/dx^0.5=2y` )

      test solution for recurrence equation to the unknown y

      by rtest( solution, eq ) or click the rtest button.

      test solution for recurrence equation to the unknown f

      by ftest( solution, eq ) or click the ftest button.
    214. bugs >>

    215. There are over 1000 bugs
    216. Tests over 1000 Differential Equations

      Transform 转换 >>

      Math handbook chapter 11 integral transform

      laplace transform

    217. First graph is in real domain, second graph is in Laplace domain by Lapalce transform
      laplace(x)

    218. Input your function, click the laplace button :
      laplace(sin(x))

      inverse laplace transform

    219. First graph is in Laplace domain , second graph is in real domain by inverse Lapalce transform
      inverselaplace(1/x^2)

      Fourier transform

    220. First graph is in real domain, second graph is in Fourier domain by Fourier transform
      fourier(x)

      Input your function, click the Fourier button :

    221. fourier(exp(x))
    222. sine wave
    223. Weierstrass function animation

      convolution transform

      First graph is in real domain, second graph is in convolution domain by convolution transform convolute(x) with x by default:
    224. Input your function, click the convolute button :
      convolute(exp(x))

    225. convoute exp(x) with 1/sqrt(x) :
      convolute(exp(x),1/sqrt(x))

      Discrete Math 离散数学 >>

      The default index variable in discrete math is k.
    226. Input harmonic(2,x), click the defintion(x) button to show its defintion, check its result by clicking the simplify(x) button, then click the limoo(x) button for its limit as x->oo.

      Difference

    227. Δ(k^2) = difference(k^2)
      Check its result by the sum(x) button

      Summation ∑

      Indefinite sum

    228. ∑ k = sum(k)

    229. Check its result by the difference(x) button
      Δ sum(k) = difference( sum(k) )

    230. In order to auto plot, the index variable should be x :
      `sum_x x` = sum(x,x)

      definite sum

    231. Definite sum = Partial sum x from 1 to x :
      1+2+ .. +x = `sum _(k=1) ^x k` = sum(k,k,1,x)

    232. Definite sum, sum x from 1 to 5 :
      1+2+ .. +5 = ∑(x,x,1,5) = sum(x,x,1,5)
      sum(x^k,k,1,5)

      Definite sum with parameter x as upper limit

      sum(k^2, k,1, x)
    233. Check its result by the difference(x) button, and then the expand(x) button.

    234. convert to sum series definition :
      tosum( exp(x) )

    235. expand above sum series by the expand(x) button :
      expand( tosum(exp(x)) )

      Indefinite sum

    236. ∑ k

    237. sum( x^k/k!,k )

    238. partial sum of 1+2+ .. + k = ∑ k = partialsum(k)

    239. Definite sum of 1+2+ .. +5 = ∑ k

      partial sum with parameter upper limit x

    240. sum(1/k^2,k,1,x)

      infinite sum

    241. sum from 1 to oo:
      Infinite sum of 1/1^2+1/2^2+1/3^2 .. +1/k^2+... = sum( 1/k^2,k,1,oo )

    242. sum from 0 to oo:
      Infinite sum of 1/0!+1/1!+1/2! .. +1/k!+... = sum( 1/k!,k,0,oo )

    243. Infinite sum x from 0 to inf :
      1/0!+1/1!+1/2!+ .. +1/x! = sum 1/(x!) as x->oo

      Series 级数

    244. convert to sum series definition :
      tosum( exp(x) ) = toseries( exp(x) )
    245. check its result by clicking the simplify(x) button :
      simplify( tosum( exp(x) ))
    246. expand above sum series :
      expand( tosum(exp(x)) )
    247. compare to Taylor series with numeric derivative:
      taylor( exp(x), x=0, 8)
    248. compare to series with symbolic derivative:
      series( exp(x) )
    249. Taylor series expansion as x=0, by default x=0.
      taylor( exp(x) as x=0 ) = taylor(exp(x))

    250. series expand not only to taylor series:
      series( exp(x) )
    251. but aslo to other series expansion:
      series( zeta(2,x) )

      the fractional order series expansion at x=0 for 5 terms and the 1.5 order

    252. series( sin(x),x,0,5,1.5 )

      Product ∏

    253. prod(x,x)

    254. `prod x`

      Definition 定义式 >>

    255. definition of function :
      definition( exp(x) )
    256. check its result by clicking the simplify(x) button :
      simplify( def(exp(x)) )
    257. series definition

    258. convert to series definition :
      toseries( exp(x) )
    259. check its result by clicking the simplify(x) button :
      simplify( tosum(exp(x)) )
    260. integral definition

    261. convert to integral definition :
      toint( exp(x) )
    262. check its result by clicking the simplify(x) button :
      simplify( toint(exp(x)) )
    263. Number Theory 数论 >>

      math handbook chapter 20 Number Theory.
      When the variable x of polynomial is numnber, it becomes polynomial number :
    264. poly number:
      poly(3,2)

    265. Hermite number:
      hermite(3,2)

    266. harmonic number:
      harmonic(-3,2)
      harmonic(-3,2,4)
      harmonic(1,1,4) = harmonic(1,4) = harmonic(4)

    267. Bell number:
      bell(5)

    268. double factorial 6!!
    269. binomial number `((4),(2))`

    270. combination number `C_2^4`

    271. harmonic number `H_4`

    272. congruence equation:
      3x-1 = 2*(mod 2)
      x^2-3x-2 = 2mod( 2)

    273. modular equation:
      Enter mod(3x,5)=1 for
      3x mod 5 = 1
      Enter mod(x^2-5x+7,2)=1 for
      (x^2-5x+7) mod 2 = 1
      Enter mod(x^2-5x+6,2)=0 for
      (x^2-5x+6) mod 2 = 0

    274. Diophantine equation:
      number of equation is less than number of the unknown, e.g. one equation with 2 unkowns x and y,
      solve( 3x-2y-2=0, x,y )
      solve( x^2-3x-2y-2=0, x,y )

      prime

    275. is prime number? isprime(12321)
    276. is prime number? is_prime(12321)
    277. Calculate the 4nd prime prime(4)
    278. next prime greater than 4 nextprime(4)
    279. next prime greater than 4 next_prime(4)
    280. prime in range 4 to 9prime_ringe(4,9)
    281. more example is number theory in function and in JavaScript javascsript math

      Probability 概率 >>

      math handbook chapter 16 Probability

    282. Probability of standard normal distribution P(range(-1,1)) between -1 and 1:
      P(range(-1,1))

    283. All probability of standard normal distribution P(x) between -oo and oo:
      P(range(-oo,oo)) = 1

    284. probability of standard normal distribution P(x<0.8):
      P(x<0.8)
    285. standard normal distribution function Phi(x):
      `Phi(x)`
    286. solve Probability equation for x by default :
      solve(P(t>x)=0.2)

    287. the binomial coefficient or choice number, combination(4,x) = C(4,2) = bimonial(4,2)
      C(4,2)

    288. arrangement number, permutation(4,x) = P(4,2),
      P(4,2)

      more example in JavaScript mathjs

      Multi elements 多元 >>

      We can put multi elements together with list(), vector(), and. Most operation in them is the same as in one element, one by one. e.g. +, -, *, /, ^, differentiation, integration, sum, etc. We count its elements with size(), as same as to count elements in function. But vector operation may be not.

      and

      Its position of the element is fix so we cannot sort it. We can plot multi curves with the and. e.g.
    289. differentiate :
      d(x and x*x)

    290. integrate :
      int(x and x*x)

      list

      There are 2 types of list: [1,2,3] is numeric list in JavaScript, and the list(1,2,3) function is symbolic list in mathHand. The symbolic list element can be symbol, formula and function. We can sort list with

      mathHand calculator on the = button

    291. two lists added value :
      list(2,3)+list(3,4)

    292. two lists added or join together
      [1,2]+[3,4]

    293. convolution of two lists
      convolution([1,2],[3,4])

      JavaScript calculation

    294. sort with JavaScript calculator:
      [1,2,3].sort()=

    295. two lists added value by the ending with =
      [1,2]+[3,4]=

    296. JavaScript calculation on the ≈ button by input:
      [1,2]+[3,4]
      click the ≈ button

      more example in JavaScript mathjs

      Statistics 统计

      math handbook chapter 16 Statistics

    297. sort(list(x)), add numbers together by total(list()), max(list()), min(list()), size(list()) with mathHand calculator on the = button. e.g.
      total(list(1,2,3))

    298. with JavaScript numeric calculator on the ≈ button :
      sum([1,2])

      more example in JavaScript mathjs

      Linear Algebra 线性代数 >>

      vector

      math handbook chapter 8 vector

      It has direction. the position of the element is fix so we cannot sort it. numeric vector is number with direction. the system auto plot the 2-dimentional vector. two vector(x) in the same dimention can be operated by +, -, *, /, and ^, the result can be checked by its reverse operation.

      There are two types of vector: symbolic vector(a,b) and numeric vector([1,2])

      symbolic vector

    299. vector(1,2)+vector(3,4)

      vector equation
    300. solve vector(1,2)+x=vector(2,4) is as same as x=vector(2,4)-vector(1,2)
    301. solve 2x-vector(2,4)=0 is as same as x=vector(2,4)/2
    302. solve 2/x-vector(2,4)=0 is as same as x=2/vector(2,4)
    303. solve vector(1,2)*x-vector(2,4)=0 is as same as x=vector(2,4)/vector(1,2)
    304. solve vector(1,2)*x-20=0 is as same as x=20/vector(1,2)
    305. solve vector(2,3)*x+vector(3,2)*y=vector(1,1),x,y is as same as solve(-1+2*x+3*y=0,-1+3*x+2*y=0)

      vector calculus
    306. differentiate vector(x,x) :
      d(vector(x,x))

    307. differentiate sin(vector(x,x)) :
      d(sin(vector(x,x)))

      numeric vector

    308. vector([1,2])+vector([3,4])

      Array 数组

      math handbook chapter 4 matrix

      Complex array [[1,2],[3,4]] can be operated by +,-,*,/,^,

    309. with array calculator 数组计算器
    310. with JavaScript numeric calculator

      more example in JavaScript mathjs

      Matrix 矩阵

      math handbook chapter 4 matrix

      Complex matrix([[1,2],[3,4]]) can be operated by +,-,*,/,\,^,

    311. with matrix calculator 矩阵计算器
    312. matrix([[1,2],[3,4]])

      programming 编程 >>

      There are many coding :
    313. math coding 数学编程
    314. HTML + JavaScript coding 网页编程
    315. cloud computing = web address coding 云计算 = 网址编程 = 网址计算器

      Graphics >>

    316. Classification by plotting function 按制图函数分类
    317. Classification by dimension 按维数分类
    318. Classification by appliaction 按应用分类
    319. Classification by objects 按物体分类
    320. Classification by function 按函数分类
    321. Classification by equation 按方程分类
    322. Classification by domain 按领域分类
    323. Classification by libary 按文库分类
    324. Classification by platform 按平台分类

      Plot 制图 >>

    325. plane curve 2D
    326. surface 2D

      3D graph 立体图 plot 3D >>

    327. space curve 3D
    328. surface 3D
    329. surface 4D

      Drawing 画画 >>

    330. drawing
    Next page => 
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