Python sympy 模块常用功能(二）

>>> limit(sin(x)/x, x, 0)
1
>>> limit(sin(x)/x, x, oo) #正无穷处极限
0
>>> limit(sin(x) * E**x, x, -oo)#负无穷处极限
0
>>> limit(1/x, x, 0, '+') #右极限
oo
>>> limit(1/x, x, 0, '-')#左极限
-oo
>>> limit(1/sin(x), x, oo) #极限不存在
AccumBounds(-oo, oo)

>>> diff(cos(x), x)
-sin(x)
>>> diff(cos(x)*sin(y), x, x, x) #多次求导（对x连续求3次偏导数）
sin(x)*sin(y)
>>> diff(cos(x)*sin(y), x,3) #与上式等价，对x求3阶偏导数
sin(x)*sin(y)
>>> diff(sinh(x)/E**y, x,x,y,y,y) 
-exp(-y)*sinh(x)
>>> diff(sinh(x)/E**y, x,2,y,3)#与上式等价。对x求2阶偏导数，再对y求3阶偏导数
-exp(-y)*sinh(x)

>>> expr = exp(x*y*z)
>>> expr.diff(x,2,y,2,z,2)
(x**4*y**4*z**4 + 12*x**3*y**3*z**3 + 38*x**2*y**2*z**2 + 32*x*y*z + 4)*exp(x*y*z)

>>> integrate(cos(x), x)
sin(x)
>>> integrate(cos(x), x,x) #对x 2 次积分
-cos(x)
>>> integrate(cos(x), x,x,x) ##对x 3 次积分
-sin(x)
>>> integrate(cos(x), x,2) #这样写会报错
SyntaxError: invalid character in identifier

>>> integrate(sinh(x)*cosh(y), x, y)
sinh(y)*cosh(x)
>>> integrate(sinh(x)*cosh(y), y, x)
sinh(y)*cosh(x)

>>> integrate(x**x, x)
Integral(x**x, x)

>>> integrate(exp(-x), (x, 0, oo)) #(integration variable, lower limit, upper limit)
1

>>> integrate(sin(x**2), x)
3*sqrt(2)*sqrt(pi)*fresnels(sqrt(2)*x/sqrt(pi))*gamma(3/4)/(8*gamma(7/4))

>>> integrate(exp(-x**2 - y**2), (x, -oo, oo), (y, -oo, oo))
pi

>>> integrate(x**y*exp(-x), (x, 0, oo))
Piecewise((gamma(y + 1), re(y) > -1), (Integral(x**y*exp(-x), (x, 0, oo)), True)) #分段函数

>>> series(sin(x)) # 默认余项次数为6，在 x=0 处展开
x - x**3/6 + x**5/120 + O(x**6)
>>> series(exp(x), x, 1, n= 3)#在 x=1 处展开，余项次数为3
E + E*(x - 1) + E*(x - 1)**2/2 + O((x - 1)**3, (x, 1))

>>> sin(x).series(x,1,n=3)
sin(1) + (x - 1)*cos(1) - (x - 1)**2*sin(1)/2 + O((x - 1)**3, (x, 1))

>>> Derivative.as_finite_difference(f(x).diff(x)) #差分近似微分
-f(x - 1/2) + f(x + 1/2)

>>>f = Function('f')
>>> diffeq = Eq(f(x).diff(x, x) - 2*f(x).diff(x) + f(x), sin(x))
>>> dsolve(diffeq, f(x))
Eq(f(x), (C1 + C2*x)*exp(x) + cos(x)/2)

>>> i, n = var("i n")
>>> summation(i, (i, 1, n)) #summation函数用于级数求和
n**2/2 + n/2
>>> summation(1/n, (n, 1, +oo)) # 调和级数，发散
oo
>>> summation(1/n**r, (n, 1, +oo)) # 仅当r>1时 收敛
Piecewise((zeta(r), r > 1), (Sum(n**(-r), (n, 1, oo)), True))

>>> from sympy.integrals import laplace_transform
>>> from sympy.abc import t, s, a #导入多个符号变量
laplace_transform(sin(t), t, s)
(1/(s**2 + 1), 0, True)
>>> laplace_transform(exp(t), t, s)
(1/(s - 1), 1, True)

>>> from sympy.integrals.transforms import inverse_laplace_transform
>>> from sympy import exp, Symbol
>>> from sympy.abc import s, t
>>> a = Symbol(’a’, positive=True)
>>> inverse_laplace_transform(exp(-a*s)/s, s, t)
Heaviside(-a + t)
>>> inverse_laplace_transform(1/(s - 1), s, t)
exp(t)*Heaviside(t)
>>> inverse_laplace_transform(s**(-2), s, t)
t*Heaviside(t)

>>> from sympy import fourier_transform, exp
>>> from sympy.abc import x, k
>>> fourier_transform(exp(-x**2), x, k)
sqrt(pi)*exp(-pi**2*k**2)
>>> fourier_transform(exp(-x**2), x, k, noconds=False)
(sqrt(pi)*exp(-pi**2*k**2), True)

>>> from sympy import inverse_fourier_transform, exp, sqrt, pi
>>> from sympy.abc import x, k
>>> inverse_fourier_transform(sqrt(pi)*exp(-(pi*k)**2), k, x)
exp(-x**2)
>>> inverse_fourier_transform(sqrt(pi)*exp(-(pi*k)**2), k, x, noconds=False)
(exp(-x**2), True)