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# Examples of Fractional Calculus Computer Algebra System 例题

## Content

• Arithmetic 算术
• Algebra 代数
• Function 函数
• Calculus 微积分
• Equation 方程
• bugs
• Transform 转换
• Discrete Math 离散数学
• Definition 定义式
• Numeric math 数值数学
• Number Theory 数论
• Probability 概率
• Multi elements 多元
1. and
2. list
• Statistics 统计
• Linear Algebra 线性代数
• programming 编程
• Graphics
• 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. Add prefix "big" to number for Big number:
big1234567890123456789
3. mod operation:
input mod(3,2) for 3 mod 2

### Complex 复数

math handbook chapter 1.1.2 complex

Complex(1,2) number is special vector, i.e. the 2-dimentional vector, so it can be operated and plotted as vector.

4. complex numbers in the complex plane:
complex(1,2) = 1+2i

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

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

7. input complex number in r*cis(theta*degree) format:
2cis(45degree)
8. #### Convert to complex(x)

9. 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) = tocomplex(1-2i)

10. input complex number in polar:
tocomplex(polar(1,45degree))

11. Convert complex a+b*i to polar(r,theta) coordinates:
convert 1-i to polar = topolar(1-i)

12. Convert complex a+b*i to polar(r,theta*degree) coordinates:
topolard(1-i)
13. 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..

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

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

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

### Numerical approximations

17. numeric computation end with the equal sign =
sin(pi/4)=

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

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

20. Convert back with numeric computation n(x) :
n(polar(2,45degree))
n( sin(pi/4) )
n( sin(30 degree) )
21. sin^((0.5))(1) is the 0.5 order derivative of sin(x) at x=1 :
n( sin(0.5,1) )
22. sin(1)^(0.5) is the 0.5 power of sin(x) at x=1 :
n( sin(1)^0.5 )
23. more are in numeric math

## Algebra 代数 >>

math handbook chapter 1 algebra

24. simplify:
taylor( (x^2 - 1)/(x-1) )
25. expand:
expand( (x-1)^3 )

26. factorization:
factor( x^4-1 )
27. factorizing:
factor( x^2+3*x+2 )
28. ### tangent

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

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

31. 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))
32. convert to exp:
toexp( cos(x) )
33. convert to trig:
convert exp(x) to trig
34. convert sin(x) to exp(x):
convert sin(x) to exp = toexp( sin(x) )

35. Convert to exp(x):
toexp(Gamma(2,x))
36. ### inverse function

37. 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

38. 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

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

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

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

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

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

45. 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 )

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

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

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

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

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

51. 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.
52. ## Function 函数 >>

math handbook chapter 1 Function 函数

#### Trigonometry 三角函数

53. expand Trigonometry by expandtrig(x) :
expandtrig( sin(x)^2 )
54. inverse function :
inverse( sin(x) )

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

56. expand trig function :
expandtrig( sin(x)^2 )
57. expand special function :
expand( gamma(2,x) )
58. factor(x) :
factor( sin(x)*cos(x) )

59. #### 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.

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

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

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

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

more are in complex function

#### special Function

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

### Limit

math handbook chapter 4.1 limit

65. 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)

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

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

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

### Derivatives

Math Handbook chapter 5 differential calculus

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

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

71. 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) )

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

73. 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

74. 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))

75. 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)

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

77. 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)

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

79. inverse the 0.5 order derivative of sin(x) function :
f(-1)( sin(0.5)(x) ) = inverse(sin(0.5,x))
80. Derive the product rule :
d/dx (f(x)*g(x)*h(x)) = d(f(x)*g(x)*h(x))
81. … as well as the quotient rule :
d/dx f(x)/g(x) = d(f(x)/g(x))
82. for derivatives :
d/dx ((sin(x)* x^2)/(1 + tan(cot(x)))) = d((sin(x)* x^2)/(1 + tan(cot(x))))
83. Multiple ways to derive functions :
d/dy cot(x*y) = d(cot(x*y) ,y)
84. Implicit derivatives, too :
d/dx (y(x)^2 - 5*sin(x)) = d(y(x)^2 - 5*sin(x))
85. the nth derivative formula :
 d^n/dx^n (sin(x)*exp(x))  = nthd(sin(x)*exp(x))
86. #### 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
87. indefinite integrate : int sin(x) dx = integrate(sin(x))

88. 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))

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

90. 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)

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

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

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

94. 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) :
95. integrator(sin(x))

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

#### fractional integrate

97. 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)

98. 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))

99. 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

100. 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.
101. ## Equation 方程 >>

equation world

### inverse an equation

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

### polynomial equation

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

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

105. 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) )

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

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

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

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

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

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

### Algebra Equation f(x)=0

math handbook chapter 3 algebaic Equation

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

113. Solve nonlinear equations:
solve(exp(x)+exp(-x)=4)

#### absolute equation

114. 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

115. solve(x) Modulus equation for the unknown x inside the mod(x) function :
input mod(x,2)=1 for
x mod 2 = 1
click the solve button

116. Enter mod(x^2-5x+7,2)=1 for
(x^2-5x+7) mod 2 = 1

117. 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

118. 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 :
x^2+3x+2=1*(mod 11)
x^2+3x+2=1 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.

119. 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)
120. solve x^2-y^2=1 graphically
x^2-y^2-1=0

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

math handbook chanpter 4.3 system of equations

121. 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 )

122. 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 )

#### 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 :
123. parametricplot( x=cos(t), y=sin(t) )
124. parametric3D( cos(t),sin(t) )
125. 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 :
126. parametric3D( cos(u*v),sin(u*v),u*v )
127. wireframe3D( cos(x*y),sin(x*y) )
128. parametric3D( cos(t),sin(t) )
129. 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 :
130. 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 :
131. parametric3D( u,u-v,u*v )
132. 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 :
133. implicit3D( x-y-z )
134. plot3D( x-y-z )

### 4D equations

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

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

### Probability_equation

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

### recurrence_equation

138. 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

139. 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

140. 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)

141. 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

142. 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

143. 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

### Inequalities

144. 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.
ODE(x) and dsolve(x) and lasove(x) solve ordinary differential equation (ODE) for unknown y.

145. Solve linear ordinary differential equations:
y'=x*y+x
y'= 2y
y'-y-1=0

146. Solve nonlinear ordinary differential equations:
(y')^2-2y^2-4y-2=0
dsolve( y' = sin(x-y) )
dsolve( y(1,x)=cos(x-y) )
dsolve( ds(y)=tan(y) )
dsolve( ds(y)=exp(y) )
dsolve( ds(y)=log(y) )

147. Solve second order nonlinear ordinary differential equations:
dsolve( ds(y,x,2)=exp(y) )
dsolve( ds(y,x,2)=log(y) )

148. 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.
149. second order ODE
y''=y'-y

### integral equation

Math handbook chapter 15 integral equation.

#### indefinite integral equation

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

#### definite integral equation

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

### differential integral equation

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

### fractional differential equation

dsolve(x) also solves fractional differential equation
153. Solve linear equations:
d^0.5/dx^0.5 y = 2y
d^0.5/dx^0.5 y -y - E_(0.5) (4x^0.5) = 0
d^0.5/dx^0.5 y -y -exp(4x) = 0
(d^0.5y)/dx^0.5=sin(x)

154. Solve nonlinear equations:
(d^0.5y)/dx^0.5 = y^2*exp(x)
(d^0.5y)/dx^0.5 = sin(y)*exp(x)
(d^0.5y)/dx^0.5 = exp(y)*exp(x)
(d^0.5y)/dx^0.5 = log(y)*exp(x)
(d^0.5y)/dx^0.5 - a*y^2-b*y-c = 0

#### fractional integral equation

155. d^-0.5/dx^-0.5 y = 2y

#### fractional differential integral equation

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

#### complex order differential equation

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

#### variable order differential equation

158. (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 :
159. linear equations:
dsolve( ds(x,t)=x-2y,ds(y,t)=2x-y )

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

161. 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 )

162. 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.

163. Solve a linear equation:
pde dy/dt = dy/dx-2y

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

#### partial differential integral equation

165. 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.

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

167. Solve nonlinear equations:
pde (d^0.5y)/dt^0.5 = 2* (d^0.5y)/dx^0.5*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

168. ds(y,t)-ints(y,x,0.5)-exp(x)=0
pde (dy)/(dt)-int y (dx)^0.5 -exp(x)=0
169. ds(y,t)-ints(y,x,0.5)+2y-exp(x)=0
pde (dy)/(dt)-int y (dx)^0.5 +2y-exp(x)=0
170. 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:
171. linear equations:
pde( ds(y,t)-ds(y,x)=2z-2y,ds(z,t)-ds(z,x)=4z-4y )

172. 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 )

173. 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 :
174. 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 :
175. test( exp(2x), dy/dx=2y )
176. 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.
177. ## bugs >>

There are over 800 bugs in wolfram software but they are no problem in MathHand.com

## Transform 转换 >>

Math handbook chapter 11 integral transform

### laplace transform

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

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

### inverse laplace transform

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

### Fourier transform

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

Input your function, click the Fourier button :

182. fourier(exp(x))
183. sine wave
184. 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:
185. Input your function, click the convolute button :
convolute(exp(x))

186. convoute exp(x) with 1/sqrt(x) :
convolute(exp(x),1/sqrt(x))
187. ## Discrete Math 离散数学 >>

The default index variable in discrete math is k.
188. 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

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

### Summation ∑

#### Indefinite sum

190. ∑ k = sum(k)

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

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

#### definite sum

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

194. 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)
195. Check its result by the difference(x) button, and then the expand(x) button.

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

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

#### Indefinite sum

198. ∑ k

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

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

201. Definite sum of 1+2+ .. +5 = ∑ k

#### partial sum with parameter upper limit x

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

#### infinite sum

203. 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 )

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

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

206. ### Series 级数

207. convert to sum series definition :
tosum( exp(x) ) = toseries( exp(x) )
208. check its result by clicking the simplify(x) button :
simplify( tosum( exp(x) ))
209. expand above sum series :
expand( tosum(exp(x)) )
210. compare to Taylor series with numeric derivative:
taylor( exp(x), x=0, 8)
211. compare to series with symbolic derivative:
series( exp(x) )
212. Taylor series expansion as x=0, by default x=0.
taylor( exp(x) as x=0 ) = taylor(exp(x))

213. series expand not only to taylor series:
series( exp(x) )
214. 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

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

### Product ∏

216. prod(x,x)

217. prod x

## Definition 定义式 >>

218. definition of function :
definition( exp(x) )
219. check its result by clicking the simplify(x) button :
simplify( def(exp(x)) )
220. #### series definition

221. convert to series definition :
toseries( exp(x) )
222. check its result by clicking the simplify(x) button :
simplify( tosum(exp(x)) )
223. #### integral definition

224. convert to integral definition :
toint( exp(x) )
225. check its result by clicking the simplify(x) button :
simplify( toint(exp(x)) )
226. ## Numeric math 数值数学 >>

math handbook chapter 3.4

227. numeric computation end with the equal sign =
sin(pi/4)=

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

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

more example in JavaScript mathjs

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

231. 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)

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

more calculus operation in JavaScript calculus

233. ## Number Theory 数论 >>

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

235. Hermite number:
hermite(3,2)

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

237. Bell number:
n(bell(5))

238. double factorial 6!!
239. Calculate the 4nd prime prime(4)
240. is prime number? isprime(12321)
241. next prime greater than 4 nextprime(4)
242. binomial number ((4),(2))

243. combination number C_2^4

244. harmonic number H_4

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

246. modular equation:
Enter mod(x-1,2)=1 for
(x-1) mod 2 = 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

247. 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 in JavaScript mathjs

248. ## Probability 概率 >>

math handbook chapter 16 Probability

249. probability of standard normal distribution P(x) :
P(x<0.8)
250. standard normal distribution function Phi(x) :
Phi(x)
251. solve Probability equation for k :
solve(P(x>k)=0.2,k)

252. combination(4,x)
combination(4,2)

more example in JavaScript mathjs

253. ## 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.
254. differentiate :
d(x and x*x)

255. 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
256. mathHand calculator on the = button :
sort(list(2,3,1))

257. JavaScript calculator on the ≈ button :
[1,2,3].sort()

more example in JavaScript mathjs

## Statistics 统计

math handbook chapter 16 Statistics

258. 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))

259. 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.

#### vector equation

260. solve vector(1,2)+x=vector(2,4) is as same as x=vector(2,4)-vector(1,2)
261. solve 2x-vector(2,4)=0 is as same as x=vector(2,4)/2
262. solve 2/x-vector(2,4)=0 is as same as x=2/vector(2,4)
263. solve vector(1,2)*x-vector(2,4)=0 is as same as x=vector(2,4)/vector(1,2)
264. solve vector(1,2)*x-20=0 is as same as x=20/vector(1,2)
265. 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

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

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

### Matrix 复数矩阵

math handbook chapter 4 matrix

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

268. with matrix calculator 矩阵计算器
269. with JavaScript numeric calculator

more example in JavaScript mathjs

## programming 编程 >>

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

## Graphics >>

273. graphics
274. Classification by plot function 按制图函数分类
275. Classification by appliaction 按应用分类

## Plot 制图 >>

276. plane curve 2D
277. surface 2D

## 3D graph 立体图 plot 3D >>

278. space curve 3D
279. surface 3D
280. surface 4D

281. drawing
• ## References

1. complex - complex math - complex function
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
9. 2D surface - 3D surface
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