# Scipy2008 examples

We couldn't go to SciPy2008, so we'll use these examples in some of our future presentation about sympy.

Besides EuroSciPy2008_examples we can also add the following:

# Numerical Integration

One-liner:

```In [1]: Integral(sin(1/x), (x, 0, 1)).transform(x, 1/x).evalf(quad="osc")
Out[1]: 0.504067061906928
```

Detailed steps:

```In [1]: e = Integral(sin(1/x), (x, 0, 1))

In [2]: e
Out[2]:
1
Ã¢Å’Â
Ã¢Å½Â®    Ã¢Å½â€º1Ã¢Å½Å¾
Ã¢Å½Â® sinÃ¢Å½Å“Ã¢â€â‚¬Ã¢Å½Å¸ dx
Ã¢Å½Â®    Ã¢Å½ÂxÃ¢Å½Â
Ã¢Å’Â¡
0

In [3]: e.transform(x, 1/x)
Out[3]:
Ã¢Ë†Å¾
Ã¢Å’Â
Ã¢Å½Â® sin(x)
Ã¢Å½Â® Ã¢â€â‚¬Ã¢â€â‚¬Ã¢â€â‚¬Ã¢â€â‚¬Ã¢â€â‚¬Ã¢â€â‚¬ dx
Ã¢Å½Â®    2
Ã¢Å½Â®   x
Ã¢Å’Â¡
1

Out[4]: 0.504067061906928

Out[5]: 0.504067061906928371989856117741148229625
```

# Numerical Summation

It works for quickly convergent series:

```>>> Sum((2*n**3+1)/factorial(2*n+1), (n, 0, oo)).evalf(1000)
1.652941212640472981900739198325231452667553042183503755040875167115365207002854
77118747045228498906167383807929789641305010501152379438610698437723585110992132
48084094702974173459412697848275449887634172363108079619463778928999727406730383
57199917316237084560028761604522443350080698146577601430156851863096927635778314
88062076063878821591479918536110213351662499708829217876455721476648748647659612
72185645529206548668821178422050797739640819097159967650626965341984007864872054
71812636349043868903125201137904072881174848578339123166638219650148561227868156
80738028532199588253087223349198266285072706513063361416254124560602074234127566
32410682925916059738774890040375938723705381947697574581499793671926177145966891
33271029543103694271529306325574205636661264488189585018019114290293809963899283
90070084916840020684307314192359067368407129281676733087681860839859648692202393
41225132757138225024317713163659365040869159437217031345698535519950979370407285
20746689993201707235774309731234398779684
```
And slowly convergent (polynomial rate) series:
```>>> Sum(n/(n**3+9), (n, 1, oo)).evalf(1000)
0.572085799521274038128017585783700438130384580104388084551740050974925897207818
98311108798290436060631856133690814143188244308005734075188518963064503611766727
51975068157408446403629166383226981406071893503958716023483643384018192761835469
62523276298459470487661766581612076405188965696292563597978253602870433142733727
49456336446570299555622044023184339325169717382623431811996989431779585758743983
22657597287758887471781904704253408614010644740045975234864559308102917760390712
09858646969081826648914656188008932364779703396061488751933093758374187906616981
59935678929938625204474297765447285426340636797285832219467575552277926359443579
66448919469783095915588358346137013995560248274612167594346431054534148807909065
87026974372235853955946903025185089032108053973102877186484901797732760077569507
62103250578219908729410121672429672442237773445952371487389948096056503557145790
85480428757289997024542130099656261002247342979582278399887560907241960471987518
890694794314366435375093779451882224094794
```

# Numerical Simplification

```In [4]: float(1/7)
Out[4]: 0.142857142857

In [5]: nsimplify(_)
Out[5]: 1/7

In [6]: float(1/81)
Out[6]: 0.0123456790123

In [7]: nsimplify(_)
Out[7]: 1/81

>>> nsimplify(pi, tolerance=0.01)
22/7
>>> nsimplify(pi, tolerance=0.001)
355/113
>>> nsimplify(0.33333, tolerance=1e-4)
1/3
>>> nsimplify(4.71, [pi], tolerance=0.01)
3*pi/2
>>> nsimplify(2.0**(1/3.), tolerance=0.001)
635/504
>>> nsimplify(2.0**(1/3.), tolerance=0.001, full=True)
2**(1/3)

>>> pprint(nsimplify(cos(atan('1/3'))))
____
3*\/ 10
--------
10

>>> pprint(nsimplify(4/(1+sqrt(5)), [GoldenRatio]))
-2 + 2*GoldenRatio

>>> pprint(nsimplify(2 + exp(2*atan('1/4')*I)))
49   8*I
-- + ---
17    17

>>> pprint(nsimplify((1/(exp(3*pi*I/5)+1))))

/         ___
/        \/ 5
1/2 - I*  /   1/4 + -----
\/            10

>>> pprint(nsimplify(I**I, [pi]))
-pi
---
2
e

>>> pprint(nsimplify(Sum(1/n**2, (n, 1, oo)), [pi]))
2
pi
---
6

>>> pprint(nsimplify(gamma('1/4')*gamma('3/4'), [pi]))
___
pi*\/ 2
```

# Curvilinear Coordinates

```\$ python examples/advanced/curvilinear_coordinates.py

Transformation: polar
Ï = Ïâ‹…cos(Ï†)
Ï† = Ïâ‹…sin(Ï†)
Jacobian:
âŽ¡cos(Ï†)  -Ïâ‹…sin(Ï†)âŽ¤
âŽ¢                 âŽ¥
âŽ£sin(Ï†)  Ïâ‹…cos(Ï†) âŽ¦
metric tensor g_{ij}:
âŽ¡1  0 âŽ¤
âŽ¢     âŽ¥
âŽ¢    2âŽ¥
âŽ£0  Ï âŽ¦
inverse metric tensor g^{ij}:
âŽ¡1  0 âŽ¤
âŽ¢     âŽ¥
âŽ¢   1 âŽ¥
âŽ¢0  â”€â”€âŽ¥
âŽ¢    2âŽ¥
âŽ£   Ï âŽ¦
det g_{ij}:
2
Ï
Laplace:
2
d               d
â”€â”€(f(Ï, Ï†))   â”€â”€â”€â”€â”€(f(Ï, Ï†))      2
dÏ            dÏ† dÏ†              d
â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€ + â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€ + â”€â”€â”€â”€â”€(f(Ï, Ï†))
Ï               2         dÏ dÏ
Ï

Transformation: cylindrical
Ï = Ïâ‹…cos(Ï†)
Ï† = Ïâ‹…sin(Ï†)
z = z
Jacobian:
âŽ¡cos(Ï†)  -Ïâ‹…sin(Ï†)  0âŽ¤
âŽ¢                    âŽ¥
âŽ¢sin(Ï†)  Ïâ‹…cos(Ï†)   0âŽ¥
âŽ¢                    âŽ¥
âŽ£  0         0      1âŽ¦
metric tensor g_{ij}:
âŽ¡1  0   0âŽ¤
âŽ¢        âŽ¥
âŽ¢    2   âŽ¥
âŽ¢0  Ï   0âŽ¥
âŽ¢        âŽ¥
âŽ£0  0   1âŽ¦
inverse metric tensor g^{ij}:
âŽ¡1  0   0âŽ¤
âŽ¢        âŽ¥
âŽ¢   1    âŽ¥
âŽ¢0  â”€â”€  0âŽ¥
âŽ¢    2   âŽ¥
âŽ¢   Ï    âŽ¥
âŽ¢        âŽ¥
âŽ£0  0   1âŽ¦
det g_{ij}:
2
Ï
Laplace:
2
d                  d
â”€â”€(f(Ï, Ï†, z))   â”€â”€â”€â”€â”€(f(Ï, Ï†, z))      2                   2
dÏ               dÏ† dÏ†                 d                   d
â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€ + â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€ + â”€â”€â”€â”€â”€(f(Ï, Ï†, z)) + â”€â”€â”€â”€â”€(f(Ï, Ï†, z))
Ï                   2          dÏ dÏ               dz dz
Ï

Transformation: spherical
Ï = Ïâ‹…cos(Ï†)â‹…sin(Î¸)
Î¸ = Ïâ‹…sin(Ï†)â‹…sin(Î¸)
Ï† = Ïâ‹…cos(Î¸)
Jacobian:
âŽ¡cos(Ï†)â‹…sin(Î¸)  Ïâ‹…cos(Ï†)â‹…cos(Î¸)  -Ïâ‹…sin(Ï†)â‹…sin(Î¸)âŽ¤
âŽ¢                                                âŽ¥
âŽ¢sin(Ï†)â‹…sin(Î¸)  Ïâ‹…cos(Î¸)â‹…sin(Ï†)  Ïâ‹…cos(Ï†)â‹…sin(Î¸) âŽ¥
âŽ¢                                                âŽ¥
âŽ£   cos(Î¸)         -Ïâ‹…sin(Î¸)            0        âŽ¦
metric tensor g_{ij}:
âŽ¡   2         2       2       2         2       2       2                                                        âŽ¤
âŽ¢cos (Î¸) + cos (Ï†)â‹…cos (Î¸)â‹…tan (Î¸) + cos (Î¸)â‹…sin (Ï†)â‹…tan (Î¸)  0                          0                       âŽ¥
âŽ¢                                                                                                                âŽ¥
âŽ¢                                                              2                                                 âŽ¥
âŽ¢                             0                               Ï                          0                       âŽ¥
âŽ¢                                                                                                                âŽ¥
âŽ¢                                                                  2    2       2       2    2       2       2   âŽ¥
âŽ£                             0                               0   Ï â‹…cos (Ï†)â‹…sin (Î¸) + Ï â‹…cos (Î¸)â‹…sin (Ï†)â‹…tan (Î¸)âŽ¦
metric tensor g_{ij} specified by hand:
âŽ¡1  0       0     âŽ¤
âŽ¢                 âŽ¥
âŽ¢    2            âŽ¥
âŽ¢0  Ï       0     âŽ¥
âŽ¢                 âŽ¥
âŽ¢        2    2   âŽ¥
âŽ£0  0   Ï â‹…sin (Î¸)âŽ¦
inverse metric tensor g^{ij}:
âŽ¡1  0       0     âŽ¤
âŽ¢                 âŽ¥
âŽ¢   1             âŽ¥
âŽ¢0  â”€â”€      0     âŽ¥
âŽ¢    2            âŽ¥
âŽ¢   Ï             âŽ¥
âŽ¢                 âŽ¥
âŽ¢           1     âŽ¥
âŽ¢0  0   â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€âŽ¥
âŽ¢        2    2   âŽ¥
âŽ£       Ï â‹…sin (Î¸)âŽ¦
det g_{ij}:
4    2
Ï â‹…sin (Î¸)
Laplace:
2                                      2
d                   d                  d                 d
â”€â”€â”€â”€â”€(f(Ï, Î¸, Ï†))   2â‹…â”€â”€(f(Ï, Î¸, Ï†))   â”€â”€â”€â”€â”€(f(Ï, Î¸, Ï†))   â”€â”€(f(Ï, Î¸, Ï†))â‹…cos(Î¸)      2
dÎ¸ dÎ¸                 dÏ               dÏ† dÏ†               dÎ¸                        d
â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€ + â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€ + â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€ + â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€ + â”€â”€â”€â”€â”€(f(Ï, Î¸, Ï†))
2                 Ï                2    2                2                dÏ dÏ
Ï                                  Ï â‹…sin (Î¸)            Ï â‹…sin(Î¸)

Transformation: rotating disk
t = t
x = xâ‹…cos(tâ‹…w) - yâ‹…sin(tâ‹…w)
y = xâ‹…sin(tâ‹…w) + yâ‹…cos(tâ‹…w)
z = z
Jacobian:
âŽ¡             1                   0          0      0âŽ¤
âŽ¢                                                    âŽ¥
âŽ¢-wâ‹…xâ‹…sin(tâ‹…w) - wâ‹…yâ‹…cos(tâ‹…w)  cos(tâ‹…w)  -sin(tâ‹…w)  0âŽ¥
âŽ¢                                                    âŽ¥
âŽ¢wâ‹…xâ‹…cos(tâ‹…w) - wâ‹…yâ‹…sin(tâ‹…w)   sin(tâ‹…w)  cos(tâ‹…w)   0âŽ¥
âŽ¢                                                    âŽ¥
âŽ£             0                   0          0      1âŽ¦
metric tensor g_{ij}:
âŽ¡     2  2    2  2              âŽ¤
âŽ¢1 + w â‹…x  + w â‹…y   -wâ‹…y  wâ‹…x  0âŽ¥
âŽ¢                               âŽ¥
âŽ¢      -wâ‹…y          1     0   0âŽ¥
âŽ¢                               âŽ¥
âŽ¢       wâ‹…x          0     1   0âŽ¥
âŽ¢                               âŽ¥
âŽ£        0           0     0   1âŽ¦
inverse metric tensor g^{ij}:
âŽ¡ 1       wâ‹…y       -wâ‹…x     0âŽ¤
âŽ¢                             âŽ¥
âŽ¢           2  2         2    âŽ¥
âŽ¢wâ‹…y   1 + w â‹…y    -xâ‹…yâ‹…w    0âŽ¥
âŽ¢                             âŽ¥
âŽ¢             2        2  2   âŽ¥
âŽ¢-wâ‹…x   -xâ‹…yâ‹…w    1 + w â‹…x   0âŽ¥
âŽ¢                             âŽ¥
âŽ£ 0        0          0      1âŽ¦
det g_{ij}:
1
Laplace:
2                                  2                          2                          2                          2
âŽ›     2  2âŽž   d                    âŽ›     2  2âŽž   d                          d                          d                          d
âŽ1 + w â‹…x âŽ â‹…â”€â”€â”€â”€â”€(f(t, x, y, z)) + âŽ1 + w â‹…y âŽ â‹…â”€â”€â”€â”€â”€(f(t, x, y, z)) + wâ‹…yâ‹…â”€â”€â”€â”€â”€(f(t, x, y, z)) + wâ‹…yâ‹…â”€â”€â”€â”€â”€(f(t, x, y, z)) - wâ‹…xâ‹…â”€â”€â”€â”€â”€(f(t, x, y, z)) - wâ‹…xâ‹…
dy dy                              dx dx                      dx dt                      dt dx                      dy dt

2                             2                             2                      2                      2
d                         2   d                         2   d                      d                      d
â”€â”€â”€â”€â”€(f(t, x, y, z)) - xâ‹…yâ‹…w â‹…â”€â”€â”€â”€â”€(f(t, x, y, z)) - xâ‹…yâ‹…w â‹…â”€â”€â”€â”€â”€(f(t, x, y, z)) + â”€â”€â”€â”€â”€(f(t, x, y, z)) + â”€â”€â”€â”€â”€(f(t, x, y, z))
dt dy                         dy dx                         dx dy                  dt dt                  dz dz

Transformation: parabolic
Ïƒ = Ïƒâ‹…Ï„
2    2
Ï„    Ïƒ
Ï„ = â”€â”€ - â”€â”€
2    2
Jacobian:
âŽ¡Ï„   ÏƒâŽ¤
âŽ¢     âŽ¥
âŽ£-Ïƒ  Ï„âŽ¦
metric tensor g_{ij}:
âŽ¡ 2    2         âŽ¤
âŽ¢Ïƒ  + Ï„      0   âŽ¥
âŽ¢                âŽ¥
âŽ¢          2    2âŽ¥
âŽ£   0     Ïƒ  + Ï„ âŽ¦
inverse metric tensor g^{ij}:
âŽ¡   1            âŽ¤
âŽ¢â”€â”€â”€â”€â”€â”€â”€     0   âŽ¥
âŽ¢ 2    2         âŽ¥
âŽ¢Ïƒ  + Ï„          âŽ¥
âŽ¢                âŽ¥
âŽ¢            1   âŽ¥
âŽ¢   0     â”€â”€â”€â”€â”€â”€â”€âŽ¥
âŽ¢          2    2âŽ¥
âŽ£         Ïƒ  + Ï„ âŽ¦
det g_{ij}:
2  2    4    4
2â‹…Ïƒ â‹…Ï„  + Ïƒ  + Ï„
Laplace:
2                2
d                d                 âŽ›     2      3âŽž d                   âŽ›     2      3âŽž d
â”€â”€â”€â”€â”€(f(Ïƒ, Ï„))   â”€â”€â”€â”€â”€(f(Ïƒ, Ï„))      âŽ4â‹…Ïƒâ‹…Ï„  + 4â‹…Ïƒ âŽ â‹…â”€â”€(f(Ïƒ, Ï„))         âŽ4â‹…Ï„â‹…Ïƒ  + 4â‹…Ï„ âŽ â‹…â”€â”€(f(Ïƒ, Ï„))
dÏƒ dÏƒ            dÏ„ dÏ„                               dÏƒ                                  dÏ„
â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€ + â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€ + â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€ + â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€
2    2           2    2       âŽ› 2    2âŽž âŽ›   2  2      4      4âŽž   âŽ› 2    2âŽž âŽ›   2  2      4      4âŽž
Ïƒ  + Ï„           Ïƒ  + Ï„        âŽÏƒ  + Ï„ âŽ â‹…âŽ4â‹…Ïƒ â‹…Ï„  + 2â‹…Ïƒ  + 2â‹…Ï„ âŽ    âŽÏƒ  + Ï„ âŽ â‹…âŽ4â‹…Ïƒ â‹…Ï„  + 2â‹…Ïƒ  + 2â‹…Ï„ âŽ

Transformation: elliptic
Î¼ = aâ‹…cos(Î½)â‹…cosh(Î¼)
Î½ = aâ‹…sin(Î½)â‹…sinh(Î¼)
Jacobian:
âŽ¡aâ‹…cos(Î½)â‹…sinh(Î¼)  -aâ‹…cosh(Î¼)â‹…sin(Î½)âŽ¤
âŽ¢                                   âŽ¥
âŽ£aâ‹…cosh(Î¼)â‹…sin(Î½)  aâ‹…cos(Î½)â‹…sinh(Î¼) âŽ¦
metric tensor g_{ij}:
âŽ¡ 2    2        2       2     2       2                                              âŽ¤
âŽ¢a â‹…cos (Î½)â‹…sinh (Î¼) + a â‹…cosh (Î¼)â‹…sin (Î½)                      0                    âŽ¥
âŽ¢                                                                                    âŽ¥
âŽ¢                                            2    2        2       2     2       2   âŽ¥
âŽ£                    0                      a â‹…cos (Î½)â‹…sinh (Î¼) + a â‹…cosh (Î¼)â‹…sin (Î½)âŽ¦
inverse metric tensor g^{ij}:
âŽ¡                     2    2        2       2     2       2
âŽ¢                    a â‹…cos (Î½)â‹…sinh (Î¼) + a â‹…cosh (Î¼)â‹…sin (Î½)
âŽ¢â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€                                          0
âŽ¢   4    2        2       2        2       4    4        4       4     4       4
âŽ¢2â‹…a â‹…cos (Î½)â‹…cosh (Î¼)â‹…sin (Î½)â‹…sinh (Î¼) + a â‹…cos (Î½)â‹…sinh (Î¼) + a â‹…cosh (Î¼)â‹…sin (Î½)
âŽ¢
âŽ¢                                                                                                         2    2        2       2     2       2
âŽ¢                                                                                                        a â‹…cos (Î½)â‹…sinh (Î¼) + a â‹…cosh (Î¼)â‹…sin (Î½)
âŽ¢                                        0                                           â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€
âŽ¢                                                                                       4    2        2       2        2       4    4        4       4
âŽ£                                                                                    2â‹…a â‹…cos (Î½)â‹…cosh (Î¼)â‹…sin (Î½)â‹…sinh (Î¼) + a â‹…cos (Î½)â‹…sinh (Î¼) + a â‹…cosh

âŽ¤
âŽ¥
âŽ¥
âŽ¥
âŽ¥
âŽ¥
âŽ¥
âŽ¥
â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€âŽ¥
4       4   âŽ¥
(Î¼)â‹…sin (Î½)âŽ¦
det g_{ij}:
4    2        2       2        2       4    4        4       4     4       4
2â‹…a â‹…cos (Î½)â‹…cosh (Î¼)â‹…sin (Î½)â‹…sinh (Î¼) + a â‹…cos (Î½)â‹…sinh (Î¼) + a â‹…cosh (Î¼)â‹…sin (Î½)
Laplace:
2                                                                                    2
âŽ› 2    2        2       2     2       2   âŽž   d                                      âŽ› 2    2        2       2     2       2   âŽž   d
âŽa â‹…cos (Î½)â‹…sinh (Î¼) + a â‹…cosh (Î¼)â‹…sin (Î½)âŽ â‹…â”€â”€â”€â”€â”€(f(Î¼, Î½))                           âŽa â‹…cos (Î½)â‹…sinh (Î¼) + a â‹…cosh (Î¼)â‹…sin (Î½)âŽ â‹…â”€â”€â”€â”€â”€(f(Î¼, Î½))
dÎ¼ dÎ¼                                                                                dÎ½ dÎ½
â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€ + â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€
4    2        2       2        2       4    4        4       4     4       4         4    2        2       2        2       4    4        4       4
2â‹…a â‹…cos (Î½)â‹…cosh (Î¼)â‹…sin (Î½)â‹…sinh (Î¼) + a â‹…cos (Î½)â‹…sinh (Î¼) + a â‹…cosh (Î¼)â‹…sin (Î½)   2â‹…a â‹…cos (Î½)â‹…cosh (Î¼)â‹…sin (Î½)â‹…sinh (Î¼) + a â‹…cos (Î½)â‹…sinh (Î¼) + a â‹…cosh

âŽ› 2    2        2       2     2       2   âŽž âŽ›   4    4        3                 4     3       4                 4    2        3       2
âŽa â‹…cos (Î½)â‹…sinh (Î¼) + a â‹…cosh (Î¼)â‹…sin (Î½)âŽ â‹…âŽ4â‹…a â‹…cos (Î½)â‹…sinh (Î¼)â‹…cosh(Î¼) + 4â‹…a â‹…cosh (Î¼)â‹…sin (Î½)â‹…sinh(Î¼) + 4â‹…a â‹…cos (Î½)â‹…cosh (Î¼)â‹…sin (Î½)â‹…s

â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€ + â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€
4       4                   âŽ›   4    2        2       2        2       4    4        4       4     4       4   âŽž âŽ›   4    2        2       2        2
(Î¼)â‹…sin (Î½)                âŽ2â‹…a â‹…cos (Î½)â‹…cosh (Î¼)â‹…sin (Î½)â‹…sinh (Î¼) + a â‹…cos (Î½)â‹…sinh (Î¼) + a â‹…cosh (Î¼)â‹…sin (Î½)âŽ â‹…âŽ4â‹…a â‹…cos (Î½)â‹…cosh (Î¼)â‹…sin (Î½)â‹…sinh (Î¼) +

4    2       2        3           âŽž d             âŽ› 2    2        2       2     2       2   âŽž âŽ›     4    3        4                4     4
inh(Î¼) + 4â‹…a â‹…cos (Î½)â‹…sin (Î½)â‹…sinh (Î¼)â‹…cosh(Î¼)âŽ â‹…â”€â”€(f(Î¼, Î½))   âŽa â‹…cos (Î½)â‹…sinh (Î¼) + a â‹…cosh (Î¼)â‹…sin (Î½)âŽ â‹…âŽ- 4â‹…a â‹…cos (Î½)â‹…sinh (Î¼)â‹…sin(Î½) + 4â‹…a â‹…cosh (Î¼)â‹…s
dÎ¼
â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€ + â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€
4    4        4         4     4       4   âŽž                             âŽ›   4    2        2       2        2       4    4        4       4     4       4
2â‹…a â‹…cos (Î½)â‹…sinh (Î¼) + 2â‹…a â‹…cosh (Î¼)â‹…sin (Î½)âŽ                              âŽ2â‹…a â‹…cos (Î½)â‹…cosh (Î¼)â‹…sin (Î½)â‹…sinh (Î¼) + a â‹…cos (Î½)â‹…sinh (Î¼) + a â‹…cosh (Î¼)â‹…sin

3                4     2       3        2                4    3        2        2          âŽž d
in (Î½)â‹…cos(Î½) - 4â‹…a â‹…cosh (Î¼)â‹…sin (Î½)â‹…sinh (Î¼)â‹…cos(Î½) + 4â‹…a â‹…cos (Î½)â‹…cosh (Î¼)â‹…sinh (Î¼)â‹…sin(Î½)âŽ â‹…â”€â”€(f(Î¼, Î½))
dÎ½
â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€â”€
âŽž âŽ›   4    2        2       2        2         4    4        4         4     4       4   âŽž
(Î½)âŽ â‹…âŽ4â‹…a â‹…cos (Î½)â‹…cosh (Î¼)â‹…sin (Î½)â‹…sinh (Î¼) + 2â‹…a â‹…cos (Î½)â‹…sinh (Î¼) + 2â‹…a â‹…cosh (Î¼)â‹…sin (Î½)âŽ
```
##### Clone this wiki locally
You can’t perform that action at this time.
Press h to open a hovercard with more details.