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# Examples of Fractional Calculus Computer Algebra System 例题

## Content

• Arithmetic 算术
• Numeric math 数值数学
• Algebra 代数
• Function 函数
• Calculus 微积分
• Equation 方程
• Transform 转换
• Discrete Math 离散数学
• Definition 定义式
• Number Theory 数论
• Probability 概率
• Multi elements 多元
• Statistics 统计
• Linear Algebra 线性代数
• 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 calculator 按计算器分类
10. 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. If something wrong, please clear meomery by clicking the AC button, then do again.

## 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. re2D plot: real curve (blue) and imag cureve (red) on real domain
re2D(x^x)

36. im2D plot: real curve (blue) and imag cureve (red) on imag domain
im2D(x^x)

37. put imagary i with variable x for plot on imag domain:
exp(i*x)

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

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

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

39. ## Numeric math 数值数学 >>

数值数学 = 计算数学
math handbook chapter 3.4

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

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

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

43. 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 )

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

more example in JavaScript mathjs

45. 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)

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

more calculus operation in JavaScript calculus

47. ## Algebra 代数 >>

math handbook chapter 1 algebra

48. simplify:
taylor( (x^2 - 1)/(x-1) )
49. expand:
expand( (x-1)^3 )

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

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

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

57. 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))
58. convert to exp:
toexp( cos(x) )
59. convert to trig:
convert exp(x) to trig
60. convert sin(x) to exp(x):
convert sin(x) to exp = toexp( sin(x) )

61. Convert to exp(x):
toexp(Gamma(2,x))
62. ### inverse function

63. 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

64. 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

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

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

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

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

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

71. 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 )

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

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

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

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

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

77. 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.
78. ## Function 函数 >>

math handbook chapter 1 Function 函数

#### Trigonometry 三角函数

79. expand Trigonometry by expandtrig(x) :
expandtrig( sin(x)^2 )
80. inverse function :
inverse( sin(x) )

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

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

85. #### 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.

86. re2D plot: real curve (blue) and imag cureve (red) on real domain
re2D(x^x)

87. im2D plot: real curve (blue) and imag cureve (red) on imag domain
im2D(x^x)

88. put imagary i with variable x for plot on imag domain:
exp(i*x)

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

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

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

## Complex

1. complex - complex function - complex math
2. complex animate(z) or complex2D(x) for phase animation in complex plane, the independent variable must be z.
3. complex plot(z) for phase and/or modulus in complex plane, the independent variable must be z.
4. plot complex(z) for phase and/or modulus in complex plane, the independent variable must be z.
5. complex2D show re2D(x) and im2D(x) for complex 2 curves of real and imag parts in real and imag domain, 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. math handbook content 2 chapter 10 complex function
2. math handbook content 3 chapter 10 complex function
3. math handbook content 4 chapter 10 complex function
4. Complex analysis
more are in complex function

#### special Function

math handbook chapter 12 special Function
92. ## Calculus 微积分 >>

### Limit

math handbook chapter 4.1 limit

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

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

95. click the limit(x) button for limit at x->oo :
limit_(x->oo) log(x)/x = limit( log(x)/x as x->inf )
= limoo( log(x)/x )

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

### Derivatives

Math Handbook chapter 5 differential calculus

97. Differentiate
d/dx sin(x) = d(sin(x))

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

99. 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) )

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

101. 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

102. 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))

103. 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)

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

105. 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)

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

107. inverse the 0.5 order derivative of sin(x) function :
f(-1)( sin(0.5)(x) ) = inverse(sin(0.5,x))
108. Derive the product rule :
d/dx (f(x)*g(x)*h(x)) = d(f(x)*g(x)*h(x))
109. … as well as the quotient rule :
d/dx f(x)/g(x) = d(f(x)/g(x))
110. for derivatives :
d/dx ((sin(x)* x^2)/(1 + tan(cot(x)))) = d((sin(x)* x^2)/(1 + tan(cot(x))))
111. Multiple ways to derive functions :
d/dy cot(x*y) = d(cot(x*y) ,y)
112. Implicit derivatives, too :
d/dx (y(x)^2 - 5*sin(x)) = d(y(x)^2 - 5*sin(x))
113. the nth derivative formula :
 d^n/dx^n (sin(x)*exp(x))  = nthd(sin(x)*exp(x))
114. #### 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
115. indefinite integrate : int sin(x) dx = integrate(sin(x))

116. 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))

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

118. 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)

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

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

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

122. 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) :
123. integrator(sin(x))

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

#### fractional integrate

125. 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)

126. 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))

127. 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

128. 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

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

130. differentiate sin(vector(x,x)) :
d(sin(vector(x,x)))
131. ## Equation 方程 >>

equation world

#### Your operation step by step

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

### inverse an equation

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

### polynomial equation

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

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

136. 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) )

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

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

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

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

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

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

144. 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.

145. 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.

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

#### absolute equation

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

#### Modulus equation

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

##### second order equation
149. solve mod(x^2-5x+7,2)=1 for
(x^2-5x+7) mod 2 = 1

150. solve 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:
151. solve 3x=1*(mod 5)
solve mod(3x,5)=mod(1,5)
##### second order equation:
152. solve x^2+3x+2=1*(mod 11)
solve x^2+3x+2=1 mod(11)
solve x^2+3x+2=mod(11)

### Probability_equation

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

### recurrence_equation

154. 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

155. 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

156. 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)

157. 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

158. 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

159. 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

see vector

### 2 variables 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.

160. 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 )

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

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

### 3D equations

#### 3 variables 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 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 )

### system of equations

math handbook chanpter 4.3 system of equations

#### system of 2 equations

166. 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 )

167. 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 )

168. 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 solver() button :
solver( 2*x+3*y-1=0, 3*x+2*y-1=0 )

#### One variable parametric equations x=f(t), y=g(t) on 2D

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

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

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 :
172. parametric3D( cos(u*v),sin(u*v),u*v )
173. wireframe3D( cos(x*y),sin(x*y) )
174. parametric3D( cos(t),sin(t) )

#### system of 3 equations

175. system of 3 equations f(x,y,z)=0, g(x,y,z)=0 h(x,y,z)=0 for 3 unknowns x and y and z by default if the unknowns omit with the solver() button :
solve([x-y-z==0,x+y+z==6,x+y==5],x,y,z )

#### one variable parametric equations x=f(t), y=f(t), z=f(t) in 3D

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 :
176. parametric3D( t,cos(t),sin(t) )

#### 2 variables parametric equations x=f(u,v), y=f(u,v), z=f(u,v) in 3D

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 :
177. parametric3D( u,u-v,u*v )
178. parametric surface

### Inequalities

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

180. 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
181. linear ordinary differential equations:
dsolve y'=x*y+x
ode y'= 2y
ode y'-y-1=0

182. 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

183. 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 )

184. 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.
185. second order ODE
odeplot y''=y'-y

### integral equation

Math handbook chapter 15 integral equation.

#### indefinite integral equation

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

187. 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
188. linear equation
ode( int int y dx -y = exp(x) )

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

#### definite integral equation

definite integral equation
190. 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

191. 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

192. 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
193. 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 - exp(x)*x = 0

194. 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

195. ode d^-0.5/dx^-0.5 y = 2y

#### nonlinear fractional integral equation

196. ode d^-0.5/dx^-0.5 y = 2y^2

#### fractional differential integral equation

197. 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

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

#### variable order differential equation

199. 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 :
200. linear equations:
dsolve( ds(x,t)=x-2y,ds(y,t)=2x-y )

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

202. 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 )

203. 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.

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

205. 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

206. 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.

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

208. 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

209. ds(y,t)-ints(y,x,0.5)-exp(x)=0
pde (dy)/(dt)-int y (dx)^0.5 -exp(x)=0
210. ds(y,t)-ints(y,x,0.5)+2y-exp(x)=0
pde (dy)/(dt)-int y (dx)^0.5 +2y-exp(x)=0
211. 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:
212. linear equations:
pde( ds(y,t)-ds(y,x)=2z-2y,ds(z,t)-ds(z,x)=4z-4y )

213. 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 )

214. 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 :
215. test( -1, x^2-5*x-6, x )
216. 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 :
217. test( exp(2x), dy/dx=2y )
218. 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.

### test equation

219. tests over 1000 Differential Equations

### bugs

220. There are over 1000 bugs
221. ## Transform 转换 >>

Math handbook chapter 11 integral transform

### laplace transform

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

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

### inverse laplace transform

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

### Fourier transform

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

Input your function, click the Fourier button :

226. fourier(exp(x))
227. sine wave
228. 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:
229. Input your function, click the convolute button :
convolute(exp(x))

230. convoute sin(x) with exp(x) :
convolute(sin(x),exp(x))

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

## Discrete Math 离散数学 >>

The default index variable in discrete math is k.
232. 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

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

### Summation ∑

#### Indefinite sum

234. ∑ k = sum(k)

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

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

#### definite sum

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

238. 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)
239. Check its result by the difference(x) button, and then the expand(x) button.

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

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

#### Indefinite sum

242. ∑ k

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

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

245. Definite sum of 1+2+ .. +5 = ∑ k

#### partial sum with parameter upper limit x

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

#### infinite sum

247. 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 )

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

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

### Series 级数

250. convert to sum series definition :
tosum( exp(x) ) = toseries( exp(x) )
251. check its result by clicking the simplify(x) button :
simplify( tosum( exp(x) ))
252. expand above sum series :
expand( tosum(exp(x)) )
253. compare to Taylor series with numeric derivative:
taylor( exp(x), x=0, 8)
254. compare to series with symbolic derivative:
series( exp(x) )
255. Taylor series expansion as x=0, by default x=0.
taylor( exp(x) as x=0 ) = taylor(exp(x))

256. series expand not only to taylor series:
series( exp(x) )
257. 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

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

### Product ∏

259. prod(x,x)

260. prod x

## Definition 定义式 >>

261. definition of function :
definition( exp(x) )
262. check its result by clicking the simplify(x) button :
simplify( def(exp(x)) )
263. #### series definition

264. convert to series definition :
toseries( exp(x) )
265. check its result by clicking the simplify(x) button :
simplify( tosum(exp(x)) )
266. #### integral definition

267. convert to integral definition :
toint( exp(x) )
268. check its result by clicking the simplify(x) button :
simplify( toint(exp(x)) )
269. ## Number Theory 数论 >>

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

271. Hermite number:
hermite(3,2)

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

273. Bell number:
bell(5)

274. double factorial 6!!
275. binomial number ((4),(2))

276. combination number C_2^4

277. harmonic number H_4

#### prime

278. is prime number? isprime(12321)
279. is prime number? is_prime(12321)
280. Calculate the 4nd prime by prime(4)
281. Calculate the 4nd prime by nth_prime(4)
282. next prime greater than 4 by nextprime(4)
283. next prime greater than 4 by next_prime(4)
284. prime in range 4 to 9 by prime_range(4,9)
285. prime in range 1 to 9 by prime_range(9)
286. #### integer equation

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

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

289. 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 )

more example is number theory in function and in JavaScript javascsript math

290. ## Probability 概率 >>

math handbook chapter 16 Probability

291. standard normal probability density function
Guassian function gauss(x)

292. standard normal distribution function gaussi(x)
Gaussian integral gaussi(x)
293. standard normal distribution function Phi(x):
Phi(x)
294. Probability of standard normal distribution P(range(-1,1)) between -1 and 1:
P(range(-1,1))

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

296. probability of standard normal distribution P(x<0.8):
P(x<0.8)
297. solve Probability equation for x by default :
solve(P(t>x)=0.2)

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

299. 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.
300. differentiate :
d(x and x*x)

301. 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.

#### symbolic list

302. sort list value :
sort(list(4,2,1))

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

#### numeric list

304. two lists join together
[1,2]+[3,4]

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

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

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

308. 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

#### symbolic list

309. 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))

#### numeric list

310. 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

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

##### vector equation
312. solve vector(1,2)+x=vector(2,4) is as same as x=vector(2,4)-vector(1,2)
313. solve 2x-vector(2,4)=0 is as same as x=vector(2,4)/2
314. solve 2/x-vector(2,4)=0 is as same as x=2/vector(2,4)
315. solve vector(1,2)*x-vector(2,4)=0 is as same as x=vector(2,4)/vector(1,2)
316. solve vector(1,2)*x-20=0 is as same as x=20/vector(1,2)
317. 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
318. differentiate vector(x,x) :
d(vector(x,x))

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

#### numeric vector

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

### Array 数组

math handbook chapter 4 matrix

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

321. with array calculator 数组计算器
322. 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 +,-,*,/,\,^,

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

## programming 编程 >>

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

## Graphics >>

328. Classification by plotting function 按制图函数分类
329. Classification by dimension 按维数分类
330. Classification by appliaction 按应用分类
331. Classification by objects 按物体分类
332. Classification by function 按函数分类
333. Classification by equation 按方程分类
334. Classification by domain 按领域分类
335. Classification by libary 按文库分类
336. Classification by calculator 按计算器分类
337. Classification by platform 按平台分类

## Plot 制图 >>

338. plane curve 2D
339. surface 2D

## 3D graph 立体图 plot 3D >>

340. space curve 3D
341. surface 3D
342. surface 4D

## Drawing 画画 >>

343. drawing
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