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Математика онлайн
Примеры, как считать в программе математика онлайн и применять функции
Сначала идет определяемая функция, потом ее описание, а потом примеры для нее.
Бледным цветом выделен код, который нужно написать в форме ввода математического выражения, а жирным выделен результат вычисления.
- integrate
-
- так считаются неопределенные и определенные интегралы в программе математика онлайн
x = Symbol('x')
print integrate(x**2 + x + 1, x)
2 3
x x
x + -- + --
2 3
x = Symbol('x')
print integrate(x**2 * exp(x) * cos(x), x)
x 2 x x 2 x
e *sin(x) x *e *sin(x) x cos(x)*e x *cos(x)*e
--------- + ------------ - x*e *sin(x) - --------- + ------------
2 2 2 2
x = Symbol('x')
print integrate(exp(-x**2)*erf(x), x)
____ 2
\/ pi *erf (x)
--------------
4
x, y = symbols('xy')
print integrate(x*y, x)
2
y*x
----
2
x = Symbol('x')
print integrate(log(x), x)
-x + x*log(x)
Определенные интегралы:
x, a = symbols('xa')
print Integral(log(x), (x, 1, a))
print ''
print integrate(log(x), (x, 1, a))
a
/
|
| log(x) dx
|
/
1
1 - a + a*log(a)
Неопределенные интегралы:
x = Symbol('x')
print integrate(x)
2
x
--
2
x = Symbol('x')
y = Symbol('y')
print integrate(x*y)
2 2
x *y
-----
4
Явное вычисление неопределенных интегралов (Три метода: left, right, midpoint) с задаваемым приближением
x = Symbol('x')
i1 = Integral(sqrt(x**3+1), (x, 2, 10))
print i1.as_sum(1, method='midpoint')
print i1.as_sum(2, method='midpoint')
print i1.as_sum(3, method='midpoint')
print i1.as_sum(4, method='midpoint')
print i1.as_sum(4, method='midpoint').n()
_____
8*\/ 217
____ ____
4*\/ 65 + 12*\/ 57
_____ ______ _______
8*\/ 217 8*\/ 3081 8*\/ 52809
--------- + ---------- + -----------
3 27 27
_____ ___ ____ ____
2*\/ 730 + 4*\/ 7 + 4*\/ 86 + 6*\/ 14
124.164447891310
- apart
-
- упрощение выражения, а именно разложение по слагаемым
x = Symbol('x')
print 1/( (x+2)*(x+1) )
print apart(1/( (x+2)*(x+1) ), x)
1
───────────────
(2 + x)*(1 + x)
1 1
───── - ─────
1 + x 2 + x
- together
-
- упрощение выражения, приведение к общему знаменателю
x = Symbol('x')
print together(1/x + 1/y + 1/z)
print together(apart((x+1)/(x-1), x), x)
print together(apart(1/( (x+2)*(x+1) ), x), x)
x*y + x*z + y*z
───────────────
x*y*z
-1 - x
──────
1 - x
1
───────────────
(2 + x)*(1 + x)
- limit
-
- предел функции
x = Symbol('x')
y = Symbol('y')
print limit(sin(x)/x, x, 0)
print limit(x, x, oo)
print limit(1/x, x, oo)
print limit(x**x, x, 0)
print limit((x+y^2)/(x+y), x, 0)
oo
0
1
y
- diff
-
- производная функции, в последних примерах производные высоких порядков - второго и третьего и т.д.
x, y = symbols('xy')
print diff(sin(x), x)
print diff(sin(2*x), x)
print diff(tan(x), x)
print limit((tan(x+y)-tan(x))/y, y, 0)
print limit((x+y^2)/(x+y), x, 0)
print diff(sin(2*x), x, 1)
print diff(sin(2*x), x, 2)
print diff(sin(2*x), x, 3)
cos(x)
2*cos(2*x)
2
1 + tan (x)
2
1 + tan (x)
y
2*cos(2*x)
-4*sin(2*x)
-8*cos(2*x)
- .series
-
- ряд Тейлора и Маклорена
x, y = symbols('xy')
print cos(x).series(x, 0, 10)
print (1/cos(x)).series(x, 0, 10)
print 1/(x + y).series(x, 0, 5)
print limit((tan(x+y)-tan(x))/y, y, 0)
print limit((x+y^2)/(x+y), x, 0)
2 4 6 8
x x x x
1 - -- + -- - --- + ----- + O(x**10)
2 24 720 40320
2 4 6 8
x 5*x 61*x 277*x
1 + -- + ---- + ----- + ------ + O(x**10)
2 24 720 8064
1
-----
x + y
2
1 + tan (x)
y
- I
-
- комплексное число
x = symbols('x')
print exp(I*x).expand()
print exp(I*x).expand(complex=True)
x = Symbol("x", real=True)
print exp(I*x).expand(complex=True)
exp(I*x)
I*exp(-im(x))*sin(re(x)) + cos(re(x))*exp(-im(x))
I*sin(x) + cos(x)
- здесь - expand - это разложить
- sin, cos, asin, acos, tan, sinh, cosh, asinh, acosh, atan
-
- Разложение тригонометрических выражений
x, y = symbols('xy')
print sin(x+y).expand(trig=True)
print cos(x+y).expand(trig=True)
print sin(I*x)
print sinh(I*x)
print asinh(I)
print asinh(I*x)
print sin(x).series(x, 0, 10)
print sinh(x).series(x, 0, 10)
print asin(x).series(x, 0, 10)
print asinh(x).series(x, 0, 10)
cos(x)*sin(y) + cos(y)*sin(x)
cos(x)*cos(y) - sin(x)*sin(y)
I*sinh(x)
I*sin(x)
pi*I
----
2
I*asin(x)
3 5 7 9
x x x x
x - -- + --- - ---- + ------ + O(x**10)
6 120 5040 362880
3 5 7 9
x x x x
x + -- + --- + ---- + ------ + O(x**10)
6 120 5040 362880
3 5 7 9
x 3*x 5*x 35*x
x + -- + ---- + ---- + ----- + O(x**10)
6 40 112 1152
3 5 7 9
x 3*x 5*x 35*x
x - -- + ---- - ---- + ----- + O(x**10)
6 40 112 1152
- Ylm, theta, phi
-
- Сферические гармоники
theta = abc.theta
phi = abc.phi
print Ylm(1, 0, theta, phi)
print Ylm(1, 1, theta, phi)
print Ylm(2, 1, theta, phi)
⎽⎽⎽
╲╱ 3 *cos(θ)
────────────
⎽⎽⎽
2*╲╱ π
⎽⎽⎽ ⅈ*φ
-╲╱ 6 *│sin(θ)│*ℯ
────────────────────
⎽⎽⎽
4*╲╱ π
⎽⎽⎽⎽ ⅈ*φ
-╲╱ 30 *│sin(θ)│*cos(θ)*ℯ
────────────────────────────
⎽⎽⎽
4*╲╱ π
- factorial, EulerGamma
-
- Факториал и гамма функция
x = Symbol('x')
y = Symbol('y', integer=True)
print factorial(x)
print factorial(y)
print factorial(x).series(x, 0, 3)
Γ(1 + x)
y!
2 2 2 2
x *EulerGamma π *x
1 - x*EulerGamma + ────────────── + ───── + O(x**3)
2 12
- zeta
-
- Дзета функция
x = Symbol('x')
print zeta(4, x)
print zeta(4, 1)
print zeta(4, 2)
print zeta(4, 3)
ζ(4, x)
4
π
──
90
4
π
-1 + ──
90
4
17 π
- ── + ──
16 90
- chebyshevt, legendre, assoc_legendre, hermite
-
- Полиномы Чебышева, Лежандра, Эрмита
x = Symbol('x')
print chebyshevt(2, x)
print chebyshevt(4, x)
print legendre(2, x)
print legendre(8, x)
print assoc_legendre(2, 1, x)
assoc_legendre(2, 2, x)
hermite(3, x)
2
-1 + 2*x
2 4
1 - 8*x + 8*x
2
3*x
-1/2 + ────
2
2 4 6 8
35 315*x 3465*x 3003*x 6435*x
─── - ────── + ─────── - ─────── + ───────
128 32 64 32 128
⎽⎽⎽⎽⎽⎽⎽⎽
╱ 2
-3*x*╲╱ 1 - x
2
3 - 3*x
3
-12*x + 8*x
- dsolve
-
- Решение дифференциальных уравнений
x = Symbol('x')
f = Function('f')
print f(x).diff(x, x) + f(x)
print dsolve(f(x).diff(x, x) + f(x), f(x))
2
d
─────(f(x)) + f(x)
dx dx
C₁*sin(x) + C₂*cos(x)
- solve
-
- Решение алгебраических уравнений
x = Symbol('x')
print solve(x**4 - 1, x)
print solve([x + 5*y - 2, -3*x + 6*y - 15], [x, y])
[ⅈ, 1, -1, -ⅈ]
{y: 1, x: -3}
- Matrix
-
- Операции с матрицами
x = Symbol('x')
y = Symbol('y')
A = Matrix([[1,x], [y,1]])
print A
print A**2
print Matrix([[1,0], [0,1]])
print Matrix(2, 3, [1, 2, 3, 4, 5, 6])
print Matrix(3, 4, lambda i,j: 1 - (i+j) % 2)
print eye(4)
print zeros(2)
print zeros((2, 5))
print ones(3)
print ones((1, 3))
M = Matrix(2, 3, [1, 2, 3, 4, 5, 6])
print M
print M[4]
print M[1,2]
print M[0,0]
print M[1,1]
print M[0:2,0:2]
print M[1:2,2]
print M[:,2]
M2 = M[:,:]
M2[0,0] = 100
print M
M = Matrix(([1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16]))
print M
M[2,2] = M[0,3] = 0
print M
M = Matrix(([1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16]))
M[2:,2:] = Matrix(2,2,lambda i,j: 0)
print M
M = Matrix(([1,2,3],[4,5,6],[7,8,9]))
print M - M
print M * M
M2 = Matrix(3,1,[1,5,0])
print M*M2
print M**2
M.row_del(0)
print M
M.col_del(1)
print M
v1 = Matrix([1,2,3])
v2 = Matrix([4,5,6])
v3 = v1.cross(v2)
print v1.dot(v2)
print v2.dot(v3)
print v1.dot(v3)
M1 = eye(3)
M2 = zeros((3, 4))
print M1.row_join(M2)
M3 = zeros((4, 3))
print M1.col_join(M3)
M = eye(3)
print 2*M
print 3*M
f = lambda x: 2*x
print eye(3).applyfunc(f)
x = Symbol('x')
M = eye(3) * x
print M
print M.subs(x, 4)
y = Symbol('y')
print M.subs(x, y)
M = Matrix(( [1, 2, 3], [3, 6, 2], [2, 0, 1] ))
print M.det()
M2 = eye(3)
print M2.det()
M3 = Matrix(( [1, 0, 0], [1, 0, 0], [1, 0, 0] ))
print M3.det()
print M2.inv()
print M2.inv('LU')
print M.inv('LU')
print M * M.inv('LU')
A = Matrix([[1,1,1],[1,1,3],[2,3,4]])
Q, R = A.QRdecomposition()
print Q*R
print Q
A = Matrix([ [2, 3, 5], [3, 6, 2], [8, 3, 6] ])
x = Matrix(3,1,[3,7,5])
b = A*x
soln = A.LUsolve(b)
print soln
L = [Matrix((2,3,5)), Matrix((3,6,2)), Matrix((8,3,6))]
out1 = GramSchmidt(L)
out2 = GramSchmidt(L, True)
print out1
print out2
M = eye(3)
M.col_del(1)
print M
M = Matrix(3,3,lambda i,j: i+j)
V = zeros((3, 1))
print V
print M.col_insert(1,V)
M = Matrix(3,3,lambda i,j: i+j)
V = Matrix(1,3,lambda i,j: 3+i+j)
print M.col_join(V)
print M.conjugate()
A = Matrix([[1, 3, 0, 0], [y, z*z, 0, 0], [0, 0, x, 0], [0, 0, 0, 0]])
a1, a2, a3 = A.get_diag_blocks()
print a1
print a2
print a3
rho = abc.rho
phi = abc.phi
X = Matrix([rho*cos(phi), rho*sin(phi), rho**2])
Y = Matrix([rho, phi])
print X.jacobian(Y)
m = Matrix(2,3,lambda i,j: 1)
print m
print m.reshape(1,6)
print m.reshape(3,2)
m = Matrix(((1,2+I),(3,4)))
print m.transpose()
print m.T
m = Matrix([ [1,3], [2,4] ])
print m.vec()
m = Matrix([ [1,2], [2,3] ])
print m.vech()
[1 x]
[ ]
[y 1]
[1 + x*y 2*x ]
[ ]
[ 2*y 1 + x*y]
[1 0]
[ ]
[0 1]
[1 2 3]
[ ]
[4 5 6]
[1 0 1 0]
[ ]
[0 1 0 1]
[ ]
[1 0 1 0]
[1 0 0 0]
[ ]
[0 1 0 0]
[ ]
[0 0 1 0]
[ ]
[0 0 0 1]
[0 0]
[ ]
[0 0]
[0 0 0 0 0]
[ ]
[0 0 0 0 0]
[1 1 1]
[ ]
[1 1 1]
[ ]
[1 1 1]
[1 1 1]
[1 2 3]
[ ]
[4 5 6]
5
6
1
5
[1 2]
[ ]
[4 5]
[6]
[3]
[ ]
[6]
[1 2 3]
[ ]
[4 5 6]
[1 2 3 4 ]
[ ]
[5 6 7 8 ]
[ ]
[9 10 11 12]
[ ]
[13 14 15 16]
[1 2 3 0 ]
[ ]
[5 6 7 8 ]
[ ]
[9 10 0 12]
[ ]
[13 14 15 16]
[1 2 3 4]
[ ]
[5 6 7 8]
[ ]
[9 10 0 0]
[ ]
[13 14 0 0]
[0 0 0]
[ ]
[0 0 0]
[ ]
[0 0 0]
[30 36 42 ]
[ ]
[66 81 96 ]
[ ]
[102 126 150]
[11]
[ ]
[29]
[ ]
[47]
[30 36 42 ]
[ ]
[66 81 96 ]
[ ]
[102 126 150]
[4 5 6]
[ ]
[7 8 9]
[4 6]
[ ]
[7 9]
32
0
0
[1 0 0 0 0 0 0]
[ ]
[0 1 0 0 0 0 0]
[ ]
[0 0 1 0 0 0 0]
[1 0 0]
[ ]
[0 1 0]
[ ]
[0 0 1]
[ ]
[0 0 0]
[ ]
[0 0 0]
[ ]
[0 0 0]
[ ]
[0 0 0]
[2 0 0]
[ ]
[0 2 0]
[ ]
[0 0 2]
[3 0 0]
[ ]
[0 3 0]
[ ]
[0 0 3]
[2 0 0]
[ ]
[0 2 0]
[ ]
[0 0 2]
[x 0 0]
[ ]
[0 x 0]
[ ]
[0 0 x]
[4 0 0]
[ ]
[0 4 0]
[ ]
[0 0 4]
[y 0 0]
[ ]
[0 y 0]
[ ]
[0 0 y]
-28
1
0
[1 0 0]
[ ]
[0 1 0]
[ ]
[0 0 1]
[1 0 0]
[ ]
[0 1 0]
[ ]
[0 0 1]
[-3/14 1/14 1/2 ]
[ ]
[-1/28 5/28 -1/4]
[ ]
[ 3/7 -1/7 0 ]
[1 0 0]
[ ]
[0 1 0]
[ ]
[0 0 1]
[1 1 1]
[ ]
[1 1 3]
[ ]
[2 3 4]
[ ___ ___ ___]
[\/ 6 -\/ 3 -\/ 2 ]
[----- ------ ------]
[ 6 3 2 ]
[ ]
[ ___ ___ ___ ]
[\/ 6 -\/ 3 \/ 2 ]
[----- ------ ----- ]
[ 6 3 2 ]
[ ]
[ ___ ___ ]
[\/ 6 \/ 3 ]
[----- ----- 0 ]
[ 3 3 ]
[3]
[ ]
[7]
[ ]
[5]
[ 23 ] [ 1692 ]
[[2], [ -- ], [ ---- ]]
[ ] [ 19 ] [ 353 ]
[3] [ ] [ ]
[ ] [ 63 ] [ 1551]
[5] [ -- ] [- ----]
[ 19 ] [ 706 ]
[ ] [ ]
[ 47] [ 423 ]
[- --] [- --- ]
[ 19] [ 706 ]
[ ____ ] [ ______ ] [ _____ ]
[ \/ 38 ] [23*\/ 6707 ] [12*\/ 706 ]
[[ ------ ], [----------- ], [---------- ]]
[ 19 ] [ 6707 ] [ 353 ]
[ ] [ ] [ ]
[ ____] [ ______ ] [ _____]
[3*\/ 38 ] [63*\/ 6707 ] [-11*\/ 706 ]
[--------] [----------- ] [-----------]
[ 38 ] [ 6707 ] [ 706 ]
[ ] [ ] [ ]
[ ____] [ ______] [ _____ ]
[5*\/ 38 ] [-47*\/ 6707 ] [-3*\/ 706 ]
[--------] [------------] [---------- ]
[ 38 ] [ 6707 ] [ 706 ]
[1 0]
[ ]
[0 0]
[ ]
[0 1]
[0]
[ ]
[0]
[ ]
[0]
[0 0 1 2]
[ ]
[1 0 2 3]
[ ]
[2 0 3 4]
[0 1 2]
[ ]
[1 2 3]
[ ]
[2 3 4]
[ ]
[3 4 5]
[0 1 2]
[ ]
[1 2 3]
[ ]
[2 3 4]
matrix support
- Rational
-
- Работа с рациональными числами
a = Rational(1,2)
print a
print a*2
print Rational(2)**50/Rational(10)**50
print pi**2
print pi.evalf()
print (pi+exp(1)).evalf()
print oo > 99999
print oo + 1
1/2
1
1/88817841970012523233890533447265625
2
pi
3.14159265358979
5.85987448204884
True
oo
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