# 微积分¶

## 区别¶

sage: var('x k w')
(x, k, w)
sage: f = x^3 * e^(k*x) * sin(w*x); f
x^3*e^(k*x)*sin(w*x)
sage: f.diff(x)
w*x^3*cos(w*x)*e^(k*x) + k*x^3*e^(k*x)*sin(w*x) + 3*x^2*e^(k*x)*sin(w*x)
sage: latex(f.diff(x))
w x^{3} \cos\left(w x\right) e^{\left(k x\right)} + k x^{3} e^{\left(k x\right)} \sin\left(w x\right) + 3 \, x^{2} e^{\left(k x\right)} \sin\left(w x\right)


var('x k w')
f = x^3 * e^(k*x) * sin(w*x)
show(f)
show(f.diff(x))


R = PolynomialRing(QQ,"x")
x = R.gen()
p = x^2 + 1
show(p.derivative())
show(p.integral())


sage: R = PolynomialRing(QQ,"x")
sage: x = R.gen()
sage: p = x^2 + 1
sage: p.derivative()
2*x
sage: p.integral()
1/3*x^3 + x


### 临界点¶

sage: x = PolynomialRing(RationalField(), 'x').gen()
sage: f1 = x^0
sage: f2 = 1-x
sage: f3 = 2*x
sage: f4 = 10*x-x^2
sage: f = piecewise([((0,1),f1), ((1,2),f2), ((2,3),f3), ((3,10),f4)])
sage: f.critical_points()
[5.0]


### 幂级数¶

Sage提供了几种构造和使用幂级数的方法。

sage: var('f0 k x')
(f0, k, x)
sage: g = f0/sinh(k*x)^4
sage: g.taylor(x, 0, 3)
-62/945*f0*k^2*x^2 + 11/45*f0 - 2/3*f0/(k^2*x^2) + f0/(k^4*x^4)


sage: (1/(2-cos(x))).series(x,7)
1 + (-1/2)*x^2 + 7/24*x^4 + (-121/720)*x^6 + Order(x^7)


sage: f = log(sin(x)/x)
sage: f.taylor(x, 0, 10)
-1/467775*x^10 - 1/37800*x^8 - 1/2835*x^6 - 1/180*x^4 - 1/6*x^2
sage: maxima(f).powerseries(x,0)._sage_()
sum(2^(2*i... - 1)*(-1)^i...*x^(2*i...)*bern(2*i...)/(i...*factorial(2*i...)), i..., 1, +Infinity)


sage: R.<w> = QQ[[]]
sage: ps = w + 17/2*w^2 + 15/4*w^4 + O(w^6); ps
w + 17/2*w^2 + 15/4*w^4 + O(w^6)
sage: ps.exp()
1 + w + 9*w^2 + 26/3*w^3 + 265/6*w^4 + 413/10*w^5 + O(w^6)
sage: (1+ps).log()
w + 8*w^2 - 49/6*w^3 - 193/8*w^4 + 301/5*w^5 + O(w^6)
sage: (ps^1000).coefficients()
[1, 8500, 36088875, 102047312625, 1729600092867375/8]


## 整合¶

Sage可以自行集成一些简单的功能：

sage: f = x^3
sage: f.integral(x)
1/4*x^4
sage: integral(x^3,x)
1/4*x^4
sage: f = x*sin(x^2)
sage: integral(f,x)
-1/2*cos(x^2)


Sage还可以计算涉及极限的符号定积分。

sage: var('x, k, w')
(x, k, w)
sage: f = x^3 * e^(k*x) * sin(w*x)
sage: f.integrate(x)
((24*k^3*w - 24*k*w^3 - (k^6*w + 3*k^4*w^3 + 3*k^2*w^5 + w^7)*x^3 + 6*(k^5*w + 2*k^3*w^3 + k*w^5)*x^2 - 6*(3*k^4*w + 2*k^2*w^3 - w^5)*x)*cos(w*x)*e^(k*x) - (6*k^4 - 36*k^2*w^2 + 6*w^4 - (k^7 + 3*k^5*w^2 + 3*k^3*w^4 + k*w^6)*x^3 + 3*(k^6 + k^4*w^2 - k^2*w^4 - w^6)*x^2 - 6*(k^5 - 2*k^3*w^2 - 3*k*w^4)*x)*e^(k*x)*sin(w*x))/(k^8 + 4*k^6*w^2 + 6*k^4*w^4 + 4*k^2*w^6 + w^8)
sage: integrate(1/x^2, x, 1, infinity)
1


### 卷积¶

sage: x = PolynomialRing(QQ, 'x').gen()
sage: f = piecewise([((0,1),1*x^0)])
sage: g = f.convolution(f)
sage: h = f.convolution(g)
sage: set_verbose(-1)
sage: P = f.plot(); Q = g.plot(rgbcolor=(1,1,0)); R = h.plot(rgbcolor=(0,1,1))


### 积分的Riemann与梯形和¶

• 计算（用于绘图）由梯形规则定义的分段线性函数，用于基于细分的数值积分 $$N$$ 子区间

• 梯形法则给出的近似值，

• 在数值积分中，根据细分到 $$N$$ 子区间，

• 由黎曼和近似给出的近似值。

sage: f1(x) = x^2
sage: f2(x) = 5-x^2
sage: f = piecewise([[[0,1], f1], [RealSet.open_closed(1,2), f2]])
sage: t = f.trapezoid(2); t
piecewise(x|-->1/2*x on (0, 1/2), x|-->3/2*x - 1/2 on (1/2, 1), x|-->7/2*x - 5/2 on (1, 3/2), x|-->-7/2*x + 8 on (3/2, 2); x)
sage: t.integral()
piecewise(x|-->1/4*x^2 on (0, 1/2), x|-->3/4*x^2 - 1/2*x + 1/8 on (1/2, 1), x|-->7/4*x^2 - 5/2*x + 9/8 on (1, 3/2), x|-->-7/4*x^2 + 8*x - 27/4 on (3/2, 2); x)
sage: t.integral(definite=True)
9/4


### 拉普拉斯变换¶

sage: var('x s')
(x, s)
sage: f1(x) = 1
sage: f2(x) = 1-x
sage: f = piecewise([((0,1),f1), ((1,2),f2)])
sage: f.laplace(x, s)
-e^(-s)/s + (s + 1)*e^(-2*s)/s^2 + 1/s - e^(-s)/s^2


sage: var('k, s, t')
(k, s, t)
sage: f = 1/exp(k*t)
sage: f.laplace(t,s)
1/(k + s)


sage: var('s, t')
(s, t)
sage: f = t^5*exp(t)*sin(t)
sage: L = laplace(f, t, s); L
3840*(s - 1)^5/(s^2 - 2*s + 2)^6 - 3840*(s - 1)^3/(s^2 - 2*s + 2)^5 +
720*(s - 1)/(s^2 - 2*s + 2)^4


## 常微分方程¶

sage:desolvers?


sage: y=function('y')(x); desolve(diff(y,x,2) + 3*x == y, dvar = y, ics = [1,1,1])
3*x - 2*e^(x - 1)
sage: desolve(diff(y,x,2) + 3*x == y, dvar = y)
_K2*e^(-x) + _K1*e^x + 3*x
sage: desolve(diff(y,x) + 3*x == y, dvar = y)
(3*(x + 1)*e^(-x) + _C)*e^x
sage: desolve(diff(y,x) + 3*x == y, dvar = y, ics = [1,1]).expand()
3*x - 5*e^(x - 1) + 3

sage: f=function('f')(x); desolve_laplace(diff(f,x,2) == 2*diff(f,x)-f, dvar = f, ics = [0,1,2])
x*e^x + e^x

sage: desolve_laplace(diff(f,x,2) == 2*diff(f,x)-f, dvar = f)
-x*e^x*f(0) + x*e^x*D[0](f)(0) + e^x*f(0)


sage: octave.de_system_plot(['x+y','x-y'], [1,-1], [0,2]) # optional - octave


$x'=x+y，x（0）=1；y'=x-y，y（0）=-1，$

sage: x, y, t = var('x y t')
sage: P=desolve_system_rk4([x+y, x-y], [x,y], ics=[0,1,-1], ivar=t, end_points=2)
sage: p1 = list_plot([[i,j] for i,j,k in P], plotjoined=True)
sage: p2 = list_plot([[i,k] for i,j,k in P], plotjoined=True, color='red')
sage: p1+p2
Graphics object consisting of 2 graphics primitives


sage: t=var('t'); x=function('x',t); y=function('y',t)
sage: des = [diff(x,t) == x+y, diff(y,t) == x-y]
sage: desolve_system(des, [x,y], ics = [0, 1, -1])
[x(t) == cosh(sqrt(2)*t), y(t) == sqrt(2)*sinh(sqrt(2)*t) - cosh(sqrt(2)*t)]


sage: R.<t> = PowerSeriesRing(QQ, default_prec=10)
sage: a = 2 - 3*t + 4*t^2 + O(t^10)
sage: b = 3 - 4*t^2 + O(t^7)
sage: f = a.solve_linear_de(prec=5, b=b, f0=3/5)
sage: f
3/5 + 21/5*t + 33/10*t^2 - 38/15*t^3 + 11/24*t^4 + O(t^5)
sage: f.derivative() - a*f - b
O(t^4)


## 周期函数的傅里叶级数¶

$$f$$ 是一个实值周期函数 2L . 傅里叶级数 f

$S(x) = \frac{a_0}{2} + \sum_{n=1}^\infty \left[a_n\cos\left(\frac{n\pi x}{L}\right) + b_n\sin\left(\frac{n\pi x}{L}\right)\right]$

$a{n=frac{1}{L}int{-L}^lf（x）cosleft（frac{npix}{L}right）dx，$

$b{n=frac{1}{L}int{L}^L f（x）sin左（frac{npi x}{L}right）dx，$

sage: f = piecewise([((0,pi/2), -1), ((pi/2,pi), 2)])
sage: f.fourier_series_cosine_coefficient(0)
1
sage: f.fourier_series_sine_coefficient(5)
-6/5/pi
sage: s5 = f.fourier_series_partial_sum(5); s5
-6/5*sin(10*x)/pi - 2*sin(6*x)/pi - 6*sin(2*x)/pi + 1/2
sage: plot(f, (0,pi)) + plot(s5, (x,0,pi), color='red')
Graphics object consisting of 2 graphics primitives