From SymPy

Contents 1 Integration 2 Linear Solving 3 Nonlinear Solving 4 Factorization 5 Term Rewriting 6 Matrices

Integration

<source lang="python"> In [1]: integrate(LambertW(x), x) Out[1]:

                         x     

-x + x⋅LambertW(x) + ───────────

                    LambertW(x)

In [2]: _.diff(x) Out[2]:

      LambertW(x)                        1                      1            

-1 + ─────────────── + LambertW(x) + ─────────── - ───────────────────────────

    1 + LambertW(x)                 LambertW(x)   (1 + LambertW(x))â‹…LambertW(

── x)

In [3]: trim(_) Out[3]: LambertW(x)

In [4]: var('a b') Out[4]: (a, b)

In [5]: integrate(a*LambertW(b*x), x) Out[5]:

 ⎛                             x      ⎞

a⋅⎜-x + x⋅LambertW(b⋅x) + ─────────────⎟

 ⎝                       LambertW(b⋅x)⎠

In [6]: _.diff(x) Out[6]:

  ⎛                      LambertW(b⋅x)           1                         1 

-a⋅⎜1 - LambertW(b⋅x) - ───────────────── - ───────────── + ──────────────────

  ⎝                    1 + LambertW(b⋅x)   LambertW(b⋅x)   (1 + LambertW(b⋅x)

              ⎞

───────────────⎟ )⋅LambertW(b⋅x)⎠

In [7]: trim(_) Out[7]: aâ‹…LambertW(bâ‹…x)

In [8]: integrate(sin(2*sqrt(x)), x) Out[8]:

  ⎛    ⎽⎽⎽⎞                     

sin⎝2⋅╲╱ x ⎠ ⎽⎽⎽ ⎛ ⎽⎽⎽⎞ ──────────── - ╲╱ x ⋅cos⎝2⋅╲╱ x ⎠

    2                           

In [9]: f = (x-tan(x))/tan(x)**2 + tan(x)

In [10]: integrate(f, x) Out[10]:

  ⎛       2   ⎞             2

log⎝1 + tan (x)⎠ x x ──────────────── - ────── - ──

      2           tan(x)   2 

</source>

Linear Solving

<source lang="python"> In [11]: var('a b c d e f g h i') Out[11]: (a, b, c, d, e, f, g, h, i)

In [12]: f_1 = a*x + b*y + c*z - 1

In [13]: f_2 = d*x + e*y + f*z - 1

In [14]: f_3 = g*x + h*y + i*z - 1

In [15]: s = solve([f_1, f_2, f_3], x, y, z)

In [16]: s Out[16]: ⎧ a⋅i + c⋅d + f⋅g - a⋅f - c⋅g - d⋅i b⋅f + c⋅h + e⋅i - ⎨y: ─────────────────────────────────────────────, x: ──────────────────────── ⎩ a⋅e⋅i + b⋅f⋅g + c⋅d⋅h - a⋅f⋅h - b⋅d⋅i - c⋅e⋅g a⋅e⋅i + b⋅f⋅g + c⋅d⋅h -

b⋅i - c⋅e - f⋅h a⋅e + b⋅g + d⋅h - a⋅h - b⋅d - e⋅g ⎫ ─────────────────────, z: ─────────────────────────────────────────────⎬ a⋅f⋅h - b⋅d⋅i - c⋅e⋅g a⋅e⋅i + b⋅f⋅g + c⋅d⋅h - a⋅f⋅h - b⋅d⋅i - c⋅e⋅g⎭

In [17]: f_1.subs(s) Out[17]:

        aâ‹…(bâ‹…f + câ‹…h + eâ‹…i - bâ‹…i - câ‹…e - fâ‹…h)           bâ‹…(aâ‹…i + câ‹…d + fâ‹…g - 

-1 + ───────────────────────────────────────────── + ─────────────────────────

    aâ‹…eâ‹…i + bâ‹…fâ‹…g + câ‹…dâ‹…h - aâ‹…fâ‹…h - bâ‹…dâ‹…i - câ‹…eâ‹…g   aâ‹…eâ‹…i + bâ‹…fâ‹…g + câ‹…dâ‹…h - a

a⋅f - c⋅g - d⋅i) c⋅(a⋅e + b⋅g + d⋅h - a⋅h - b⋅d - e⋅g) ──────────────────── + ───────────────────────────────────────────── ⋅f⋅h - b⋅d⋅i - c⋅e⋅g a⋅e⋅i + b⋅f⋅g + c⋅d⋅h - a⋅f⋅h - b⋅d⋅i - c⋅e⋅g

In [18]: simplify(f_1.subs(s)) Out[18]: 0

In [19]: simplify(f_2.subs(s)) Out[19]: 0

In [20]: simplify(f_3.subs(s)) Out[20]: 0 </source>

Nonlinear Solving

<source lang="python"> In [21]: f_1 = x**2 + y + z - 1

In [22]: f_2 = x + y**2 + z - 1

In [23]: f_3 = x + y + z**2 - 1

In [24]: solve([f_1, f_2, f_3], x, y, z) Out[24]: ⎡⎛ ⎽⎽⎽ ⎽⎽⎽ ⎽⎽⎽⎞ ⎛ ⎽⎽⎽ ⎣⎝-1 - ╲╱ 2 , -1 - ╲╱ 2 , -1 - ╲╱ 2 ⎠, (0, 0, 1), (0, 1, 0), ⎝-1 + ╲╱ 2 , -1 +

  ⎽⎽⎽         ⎽⎽⎽⎞           ⎤
╲╱ 2 , -1 + ╲╱ 2 ⎠, (1, 0, 0)⎦

</source>

Factorization

<source lang="python"> In [1]: factor(x**56-1, x) Out[1]:

                ⎛     2⎞ ⎛     4⎞ ⎛         2    3    4    5    6⎞ ⎛         

-(1 + x)⋅(1 - x)⋅⎝1 + x ⎠⋅⎝1 + x ⎠⋅⎝1 + x + x + x + x + x + x ⎠⋅⎝1 - x + x

2 3 4 5 6⎞ ⎛ 2 4 6 8 10 12⎞ ⎛ 4 8 12

 - x  + x  - x  + x ⎠⋅⎝1 - x  + x  - x  + x  - x   + x  ⎠⋅⎝1 - x  + x  - x   

  16    20    24⎞

+ x - x + x ⎠

In [2]: f = x**16+4*x**15+14*x**14+32*x**13+47*x**12+92*x**11+66*x**10+120*x**9 \

  ...:     -50*x**8-24*x**7-132*x**6-40*x**5-52*x**4-64*x**3-64*x**2-32*x+16

In [3]: f Out[3]:

               2       3       4       5        6       7       8        9   

16 - 32â‹…x - 64â‹…x - 64â‹…x - 52â‹…x - 40â‹…x - 132â‹…x - 24â‹…x - 50â‹…x + 120â‹…x +

   10       11       12       13       14      15    16

66â‹…x + 92â‹…x + 47â‹…x + 32â‹…x + 14â‹…x + 4â‹…x + x

In [4]: factor(f, x) Out[4]: ⎛ 2 4 6 8⎞ ⎛ 2 3 4 5 6 ⎝4 + 4⋅x + 9⋅x + 4⋅x + x ⎠⋅⎝4 - 8⋅x - 20⋅x - 8⋅x - 2⋅x + 16⋅x + 10⋅x +

   7    8⎞
4⋅x  + x ⎠

</source>

Term Rewriting

<source lang="python"> In [29]: 36/(x**5-2*x**4-2*x**3+4*x**2+x-2) Out[29]:

              36               

────────────────────────────────

           2      3      4    5

-2 + x + 4â‹…x - 2â‹…x - 2â‹…x + x

In [30]: f = apart(36/(x**5-2*x**4-2*x**3+4*x**2+x-2), x)

In [31]: f Out[31]:

   4       4         9          3    

- ───── + ────── - ──────── - ────────

 1 + x   -2 + x          2          2
                  (1 - x)    (1 + x) 

In [32]: f = apart(36/(x**5-2*x**4-2*x**3+4*x**2+x-2), x, evaluate=False)

In [33]: f Out[33]: RootSum(Lambda(_a, 4/(x - _a)), x - 2, x) + RootSum(Lambda(_a, -1/(x - _a)**2* (6 + 3*_a)), x**2 - 1, x) + RootSum(Lambda(_a, -4/(x - _a)), x + 1, x)

In [34]: Add(*[ g.doit() for g in f.args ]) Out[34]:

   4       4         9          3    

- ───── + ────── - ──────── - ────────

 1 + x   -2 + x          2          2
                  (1 - x)    (1 + x) 

</source>

Matrices

<source lang="python"> In [35]: M = Matrix(4, 4, lambda i,j: i*j+x)

In [36]: M Out[36]: ⎡x x x x ⎤ ⎢ ⎥ ⎢x 1 + x 2 + x 3 + x⎥ ⎢ ⎥ ⎢x 2 + x 4 + x 6 + x⎥ ⎢ ⎥ ⎣x 3 + x 6 + x 9 + x⎦

In [37]: M.eigenvals() Out[37]: ⎧ ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽ ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽ ⎪ ╱ 2 ╱ 2 ⎨ ╲╱ 196 + 32⋅x + 16⋅x ╲╱ 196 + 32⋅x + 16⋅x ⎪7 + 2⋅x + ───────────────────────: 1, 7 + 2⋅x - ───────────────────────: 1, 0 ⎩ 2 2

  ⎫
  ⎪
  ⎬

2⎪

  ⎭

</source>