mpmath

https://mpmath.org/doc/current/
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### Install mpmath from source

Source: https://mpmath.org/doc/current/setup.html

Install mpmath by downloading the source package, extracting it, and running the setup script.

```bash
python setup.py install
```

--------------------------------

### Install mpmath using easy_install

Source: https://mpmath.org/doc/current/_sources/setup.rst.txt

Install mpmath using easy_install if setuptools is available. Use the -U flag to force an upgrade.

```bash
easy_install mpmath
```

```bash
python -m easy_install mpmath
```

--------------------------------

### Initialize mpmath settings

Source: https://mpmath.org/doc/current/functions/hypergeometric.html

Sets the decimal precision and pretty-printing for mpmath. This is a common setup for examples.

```python
from mpmath import *
mp.dps = 25; mp.pretty = True
a = mpf(0.75)
b = mpf(1.5)
z = mpf(2.25)
```

--------------------------------

### Install mpmath on Debian/Ubuntu

Source: https://mpmath.org/doc/current/_sources/setup.rst.txt

Install mpmath on Debian or Ubuntu systems using the apt-get package manager.

```bash
sudo apt-get install python-mpmath
```

--------------------------------

### Verify mpmath installation and basic usage

Source: https://mpmath.org/doc/current/_sources/setup.rst.txt

After installation, verify mpmath works by importing it, setting decimal places, and performing calculations.

```python
from mpmath import *
mp.dps = 50
print(mpf(2) ** mpf('0.5'))
```

```python
print(2*pi)
```

--------------------------------

### Install mpmath using pip

Source: https://mpmath.org/doc/current/_sources/setup.rst.txt

Use pip to install the latest release of mpmath. For specific versions, append '==version_number'.

```bash
pip install mpmath
```

```bash
pip install mpmath==0.19
```

--------------------------------

### Example: hyper2d() with closed-form series

Source: https://mpmath.org/doc/current/functions/hypergeometric.html

Shows an example of hyper2d() that can be evaluated in closed form. Requires mpmath to be imported.

```python
>>> hyper2d({'m':1,'n':1},{'m+n':1},x,y)
2.013417124712514809623881
>>> (exp(x)*x-exp(y)*y)/(x-y)
2.013417124712514809623881
```

--------------------------------

### Verify mpmath installation

Source: https://mpmath.org/doc/current/setup.html

Check if the mpmath library is installed and working correctly by performing calculations in the Python interpreter.

```python
>>> from mpmath import *
>>> mp.dps = 50
>>> print(mpf(2) ** mpf('0.5'))
1.4142135623730950488016887242096980785696718753769
>>> print(2*pi)
6.2831853071795864769252867665590057683943387987502
```

--------------------------------

### Incomplete Elliptic Integral Pi(n,z,m) Examples

Source: https://mpmath.org/doc/current/functions/elliptic.html

Provides examples for basic values, limits, and comparisons with other functions for the incomplete elliptic integral of the third kind. Supports complex arguments and quasi-periodic extension.

```python
>>> ellippi(0.25,-0.5); ellippi(0.25,pi/2,-0.5)
1.622944760954741603710555
1.622944760954741603710555
>>> ellippi(1,0,1)
0.0
>>> ellippi(inf,0,1)
0.0
>>> ellippi(0,0.25,0.5); ellipf(0.25,0.5)
0.2513040086544925794134591
0.2513040086544925794134591
>>> ellippi(1,1,1); (log(sec(1)+tan(1))+sec(1)*tan(1))/2
2.054332933256248668692452
2.054332933256248668692452
>>> ellippi(0.25, 53*pi/2, 0.75); 53*ellippi(0.25,0.75)
135.240868757890840755058
135.240868757890840755058
>>> ellippi(0.5,pi/4,0.5); 2*ellipe(pi/4,0.5)-1/sqrt(3)
0.9190227391656969903987269
0.9190227391656969903987269
```

```python
>>> ellippi(0.5, 5+6j-2*pi, -7-8j)
(-0.3612856620076747660410167 + 0.5217735339984807829755815j)
```

```python
>>> ellippi(1,1)
+inf
>>> ellippi(1,0)
+inf
>>> ellippi(1,2,0)
+inf
>>> ellippi(1,2,1)
+inf
>>> ellippi(1,0,1)
0.0
```

--------------------------------

### Compute Logarithmic Integrals

Source: https://mpmath.org/doc/current/functions/expintegrals.html

Examples demonstrating basic values, limits, complex arguments, and the offset logarithmic integral.

```python
>>> from mpmath import *
>>> mp.dps = 30; mp.pretty = True
>>> li(0)
0.0
>>> li(1)
-inf
>>> li(1)
-inf
>>> li(2)
1.04516378011749278484458888919
>>> findroot(li, 2)
1.45136923488338105028396848589
>>> li(inf)
+inf
>>> li(2, offset=True)
0.0
>>> li(1, offset=True)
-inf
>>> li(0, offset=True)
-1.04516378011749278484458888919
>>> li(10, offset=True)
5.12043572466980515267839286347
```

```python
>>> mp.dps = 20
>>> li(3+4j)
(3.1343755504645775265 + 2.6769247817778742392j)
```

```python
>>> ei(log(3))
2.1635885946671919729
>>> li(3)
2.1635885946671919729
```

```python
>>> mp.dps = 15
>>> x = 10**100
>>> x/log(x)
4.34294481903252e+97
>>> li(x)
4.3619719871407e+97
```

```python
>>> li(10**23, offset=True)
1.92532039161405e+21
```

```python
>>> quad(li, [0, 1])
-0.693147180559945
>>> -ln(2)
-0.693147180559945
```

--------------------------------

### Basic AiryBi function examples

Source: https://mpmath.org/doc/current/functions/bessel.html

Demonstrates basic evaluation of the `airybi` function for real and complex arguments, including limits.

```python
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> airybi(0); 1/(power(3,'1/6')*gamma('2/3'))
0.6149266274460007351509224
0.6149266274460007351509224
>>> airybi(1)
1.207423594952871259436379
>>> airybi(-1)
0.10399738949694461188869
>>> airybi(inf); airybi(-inf)
+inf
0.0
```

```python
>>> airybi(-100)
0.02427388768016013160566747
>>> airybi(100)
6.041223996670201399005265e+288
>>> airybi(50+50j)
(-5.322076267321435669290334e+63 + 1.478450291165243789749427e+65j)
>>> airybi(-50+50j)
(-3.347525544923600321838281e+157 + 1.041242537363167632587245e+158j)
```

```python
>>> airybi(10**10)
1.369385787943539818688433e+289529654602165
>>> airybi(-10**10)
0.001775656141692932747610973
>>> w = airybi(10**10*(1+j))
>>> w.real
-6.559955931096196875845858e+186339621747689
>>> w.imag
-6.822462726981357180929024e+186339621747690
```

--------------------------------

### Basic factorial usage and limits

Source: https://mpmath.org/doc/current/functions/gamma.html

Examples of factorial values, limits, and Stirling's approximation.

```python
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> for k in range(6):
...     print("%s %s" % (k, fac(k)))
...
0 1.0
1 1.0
2 2.0
3 6.0
4 24.0
5 120.0
>>> fac(inf)
+inf
>>> fac(0.5), sqrt(pi)/2
(0.886226925452758, 0.886226925452758)
```

```python
>>> x = 10**10
>>> fac(x)
2.32579620567308e+95657055186
>>> sqrt(2*pi*x)*(x/e)**x
2.32579597597705e+95657055186
```

```python
>>> fac(10**30)
6.22311232304258e+29565705518096748172348871081098
```

```python
>>> nsum(lambda k: 1/fac(k), [0, inf]), exp(1)
(2.71828182845905, 2.71828182845905)
>>> nsum(lambda k: pi**k/fac(k), [0, inf]), exp(pi)
(23.1406926327793, 23.1406926327793)
```

--------------------------------

### Integrate over real line and half-infinite intervals

Source: https://mpmath.org/doc/current/calculus/integration.html

Examples of integrating functions over (-inf, inf) and (-inf, -1).

```python
>>> quadosc(lambda x: cos(x)/(1+x**2), [-inf, inf], omega=1)
1.15572734979092
>>> pi/e
1.15572734979092
>>> quadosc(lambda x: cos(x)/x**2, [-inf, -1], period=2*pi)
-0.0844109505595739
>>> cos(1)+si(1)-pi/2
-0.0844109505595738
```

--------------------------------

### Run mpmath unit tests

Source: https://mpmath.org/doc/current/_sources/setup.rst.txt

Execute the full suite of mpmath unit tests using py.test. Ensure py.test is installed.

```bash
py.test --pyargs mpmath
```

--------------------------------

### Compute 1D integrals with infinite ranges

Source: https://mpmath.org/doc/current/calculus/integration.html

Examples of evaluating integrals over infinite or half-infinite intervals.

```python
>>> mp.dps = 15
>>> quad(lambda x: 2/(x**2+1), [0, inf])
3.14159265358979
>>> quad(lambda x: exp(-x**2), [-inf, inf])**2
3.14159265358979
```

--------------------------------

### Compute Sine Integrals

Source: https://mpmath.org/doc/current/functions/expintegrals.html

Examples for calculating the sine integral, including limits, complex plane evaluation, and series summation.

```python
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> si(0)
0.0
>>> si(1)
0.9460830703671830149413533
>>> si(-1)
-0.9460830703671830149413533
>>> si(pi)
1.851937051982466170361053
>>> si(inf)
1.570796326794896619231322
>>> si(-inf)
-1.570796326794896619231322
>>> si(2+3j)
(4.547513889562289219853204 + 1.399196580646054789459839j)
```

```python
>>> si(10**10)
1.570796326707584656968511
>>> pi/2
1.570796326794896619231322
```

```python
>>> si(10**6*(1+j))
(-9.75744874290013526417059e+434287 + 4.449410587611035724984376e+434287j)
```

```python
>>> mp.dps = 15
>>> quad(sinc, [0, 5])
1.54993124494467
>>> si(5)
1.54993124494467
```

```python
>>> nsum(lambda k: (-1)**k/(fac(2*k+1)*(2*k+1)), [0,inf])
0.946083070367183
>>> si(1)
0.946083070367183
```

--------------------------------

### Compute Cosine Integrals

Source: https://mpmath.org/doc/current/functions/expintegrals.html

Examples for calculating the cosine integral, including limits, complex plane evaluation, and series summation.

```python
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> ci(0)
-inf
>>> ci(1)
0.3374039229009681346626462
>>> ci(pi)
0.07366791204642548599010096
>>> ci(inf)
0.0
>>> ci(-inf)
(0.0 + 3.141592653589793238462643j)
>>> ci(2+3j)
(1.408292501520849518759125 - 2.983617742029605093121118j)
```

```python
>>> ci(10**10)
-4.875060251748226537857298e-11
>>> sinc(10**10)
-4.875060250875106915277943e-11
>>> chop(limit(ci, inf))
0.0
```

```python
>>> findroot(ci, 1)
0.6165054856207162337971104
>>> findroot(ci, 2)
3.384180422551186426397851
```

```python
>>> ci(10**6*(1+j))
(4.449410587611035724984376e+434287 + 9.75744874290013526417059e+434287j)
```

```python
>>> mp.dps = 15
>>> -quadosc(lambda t: cos(t)/t, [5, inf], omega=1)
-0.190029749656644
>>> ci(5)
-0.190029749656644
```

```python
>>> nsum(lambda k: (-1)**k/(fac(2*k)*(2*k)), [1,inf])
-0.239811742000565
>>> ci(1) - euler
-0.239811742000565
```

--------------------------------

### Check mpmath Backend in Sage

Source: https://mpmath.org/doc/current/_sources/setup.rst.txt

Verify that mpmath is configured to use Sage's backend for optimized arithmetic. This is the default behavior when mpmath is installed within Sage.

```python
sage: mpmath.libmp.BACKEND
'sage'
```

--------------------------------

### Creating mpf instances

Source: https://mpmath.org/doc/current/_sources/basics.rst.txt

Shows how to instantiate mpf objects from various data types.

```python
>>> mpf(4)
mpf('4.0')
>>> mpf(2.5)
mpf('2.5')
>>> mpf("1.25e6")
mpf('1250000.0')
>>> mpf(mpf(2))
mpf('2.0')
>>> mpf("inf")
mpf('+inf')
```

--------------------------------

### Plotting Multiple Functions with mpmath.plot

Source: https://mpmath.org/doc/current/_sources/plotting.rst.txt

Use `mpmath.plot` to generate x-y graphs of functions. Ensure matplotlib is installed. This example plots the cosine and sine functions over the interval [-4, 4].

```python
plot([cos, sin], [-4, 4])
```

--------------------------------

### Creating Matrices

Source: https://mpmath.org/doc/current/matrices.html

Demonstrates how to create matrices using the `matrix` class, specifying dimensions, and initializing with zeros.

```APIDOC
## Creating Matrices

### Basic methods

Matrices in mpmath are implemented using dictionaries, efficiently storing only non-zero values for sparse matrix representation.

**Creating an empty matrix:**

```python
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = False
>>> matrix(2)
matrix(
[['0.0', '0.0'],
 ['0.0', '0.0']])
>>> matrix(2, 3)
matrix(
[['0.0', '0.0', '0.0'],
 ['0.0', '0.0', '0.0']])
```

Calling `matrix` with one dimension creates a square matrix.

**Accessing dimensions:**

```python
>>> A = matrix(3, 2)
>>> A.rows
3
>>> A.cols
2
```

**Changing dimensions:**

Resizing a matrix sets new elements to 0. Elements outside the new dimensions are discarded.

```python
>>> A.rows = 2
>>> A
matrix(
[['0.0', '0.0'],
 ['0.0', '0.0']])
```

**Setting elements:**

Elements are accessed using `A[row, column]` (0-indexed).

```python
>>> A = matrix(2)
>>> A[1,1] = 1 + 1j
>>> print(A)
[0.0           0.0]
[0.0  (1.0 + 1.0j)]
```

**Creating from nested lists:**

A more convenient way to create matrices.

```python
>>> matrix([[1, 2], [3, 4]])
matrix(
[['1.0', '2.0'],
 ['3.0', '4.0']])
```
```

--------------------------------

### Create numbers and matrices

Source: https://mpmath.org/doc/current/_sources/contexts.rst.txt

Demonstrates creating real numbers, complex numbers, and matrices using different contexts.

```python
>>> from mpmath import mp, fp
>>> mp.mpf(3)
mpf('3.0')
>>> fp.mpf(3)
3.0
```

```python
>>> mp.mpc(2,3)
mpc(real='2.0', imag='3.0')
>>> fp.mpc(2,3)
(2+3j)
```

```python
>>> mp.matrix([[1,0],[0,1]])
matrix(
[['1.0', '0.0'],
 ['0.0', '1.0']])
>>> _[0,0]
mpf('1.0')
>>> fp.matrix([[1,0],[0,1]])
matrix(
[['1.0', '0.0'],
 ['0.0', '1.0']])
>>> _[0,0]
1.0
```

--------------------------------

### Example: hyper2d() with separable cases

Source: https://mpmath.org/doc/current/functions/hypergeometric.html

Demonstrates two separable cases for hyper2d(): a product of two geometric series and a product of two Gaussian hypergeometric functions. Requires mpmath to be imported.

```python
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> x, y = mpf(0.25), mpf(0.5)
>>> hyper2d({'m':1,'n':1}, {}, x,y)
2.666666666666666666666667
>>> 1/(1-x)/(1-y)
2.666666666666666666666667
>>> hyper2d({'m':[1,2],'n':[3,4]}, {'m':[5],'n':[6]}, x,y)
4.164358531238938319669856
>>> hyp2f1(1,2,5,x)*hyp2f1(3,4,6,y)
4.164358531238938319669856
```

--------------------------------

### Create PDF documentation

Source: https://mpmath.org/doc/current/_sources/setup.rst.txt

Generate a PDF version of the mpmath documentation from the built LaTeX files.

```bash
make -C build/sphinx/latex all-pdf
```

--------------------------------

### Creating matrices with mp and fp contexts

Source: https://mpmath.org/doc/current/contexts.html

Demonstrates matrix creation using the `matrix` method from both `mp` and `fp` contexts, and accessing elements.

```python
>>> mp.matrix([[1,0],[0,1]])
matrix(
[['1.0', '0.0'],
 ['0.0', '1.0']])
```

```python
>>> _[0,0]
mpf('1.0')
```

```python
>>> fp.matrix([[1,0],[0,1]])
matrix(
[['1.0', '0.0'],
 ['0.0', '1.0']])
```

```python
>>> _[0,0]
1.0
```

--------------------------------

### Build mpmath documentation

Source: https://mpmath.org/doc/current/_sources/setup.rst.txt

Compile the mpmath documentation into HTML and LaTeX formats using Sphinx.

```bash
python setup.py build_sphinx -c docs -b html,latex
```

--------------------------------

### Get Nearest Integer and Log-Precision with nint_distance()

Source: https://mpmath.org/doc/current/general.html

Use nint_distance(x) to get a tuple (n, d), where n is the nearest integer to x and d is an estimate of log2(|x-n|). A negative d indicates precision lost due to cancellation when computing x-n.

```python
>>> from mpmath import *
>>> n, d = nint_distance(5)
>>> print(n); print(d)
5
-inf
```

```python
>>> n, d = nint_distance(mpf(5))
>>> print(n); print(d)
5
-inf
```

```python
>>> n, d = nint_distance(mpf(5.00000001))
>>> print(n); print(d)
5
-26
```

```python
>>> n, d = nint_distance(mpf(4.99999999))
>>> print(n); print(d)
5
-26
```

```python
>>> n, d = nint_distance(mpc(5,10))
>>> print(n); print(d)
5
4
```

```python
>>> n, d = nint_distance(mpc(5,0.000001))
>>> print(n); print(d)
5
-19
```

--------------------------------

### GET /mpmath/spherharm

Source: https://mpmath.org/doc/current/_sources/functions/orthogonal.rst.txt

Calculates the spherical harmonic function.

```APIDOC
## GET /mpmath/spherharm

### Description
Computes the spherical harmonic Y_l^m(theta, phi).

### Method
GET

### Parameters
#### Query Parameters
- **l** (int) - Required - Degree
- **m** (int) - Required - Order
- **theta** (float) - Required - Polar angle
- **phi** (float) - Required - Azimuthal angle
```

--------------------------------

### Complete Elliptic Integral E(z,m) Examples

Source: https://mpmath.org/doc/current/functions/elliptic.html

Demonstrates basic values, limits, and verification of the complete elliptic integral of the second kind. Supports arbitrary complex m.

```python
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> ellipe(0)
1.570796326794896619231322
>>> ellipe(1)
1.0
>>> ellipe(-1)
1.910098894513856008952381
>>> ellipe(2)
(0.5990701173677961037199612 + 0.5990701173677961037199612j)
>>> ellipe(inf)
(0.0 + +infj)
>>> ellipe(-inf)
+inf
```

```python
>>> ellipe(0.5)
1.350643881047675502520175
>>> quad(lambda t: sqrt(1-0.5*sin(t)**2), [0, pi/2])
1.350643881047675502520175
>>> pi/2*hyp2f1(0.5,-0.5,1,0.5)
1.350643881047675502520175
```

```python
>>> ellipe(0.5+0.25j)
(1.360868682163129682716687 - 0.1238733442561786843557315j)
>>> ellipe(3+4j)
(1.499553520933346954333612 - 1.577879007912758274533309j)
```

```python
>>> quad(ellipe, [0,1])
1.333333333333333333333333
```

--------------------------------

### GET /root

Source: https://mpmath.org/doc/current/functions/powers.html

Computes the n-th root of a complex number z.

```APIDOC
## GET /root

### Description
Computes an n-th root of z, returning a number r such that r^n = z. Every complex number z != 0 has n distinct n-th roots.

### Parameters
#### Query Parameters
- **z** (number/complex) - Required - The number to compute the root of.
- **n** (integer) - Required - The degree of the root.
- **k** (integer) - Optional - The index of the root to return (default is 0). Roots are indexed counterclockwise starting from the positive real half-axis.
```

--------------------------------

### GET /mpmath/legendre

Source: https://mpmath.org/doc/current/_sources/functions/orthogonal.rst.txt

Calculates the Legendre polynomial of degree n at x.

```APIDOC
## GET /mpmath/legendre

### Description
Computes the Legendre polynomial P_n(x).

### Method
GET

### Parameters
#### Query Parameters
- **n** (int) - Required - The degree of the polynomial
- **x** (float/complex) - Required - The point at which to evaluate
```

--------------------------------

### Complete Elliptic Integral Pi(n,m) Examples

Source: https://mpmath.org/doc/current/functions/elliptic.html

Shows basic values, limits, and evaluations in terms of simpler functions for the complete elliptic integral of the third kind. Handles infinite and complex arguments.

```python
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> ellippi(0,-5); ellipk(-5)
0.9555039270640439337379334
0.9555039270640439337379334
>>> ellippi(inf,2)
0.0
>>> ellippi(2,inf)
0.0
>>> abs(ellippi(1,5))
+inf
>>> abs(ellippi(0.25,1))
+inf
```

```python
>>> ellippi(0.25,0.25); ellipe(0.25)/(1-0.25)
1.956616279119236207279727
1.956616279119236207279727
>>> ellippi(3,0); pi/(2*sqrt(-2))
(0.0 - 1.11072073453959156175397j)
(0.0 - 1.11072073453959156175397j)
>>> ellippi(-3,0); pi/(2*sqrt(4))
0.7853981633974483096156609
0.7853981633974483096156609
```

--------------------------------

### Compute complex contour integrals

Source: https://mpmath.org/doc/current/calculus/integration.html

Example of integrating along a complex path.

```python
>>> mp.dps = 15
>>> chop(quad(lambda z: 1/z, [1,j,-1,-j,1]))
(0.0 + 6.28318530717959j)
```

--------------------------------

### Compute exact high order roots

Source: https://mpmath.org/doc/current/functions/powers.html

Examples of calculating roots for large powers.

```python
>>> root(75**210, 105)
5625.0
>>> root(1, 128, 96)
(0.0 - 1.0j)
>>> root(4**128, 128, 96)
(0.0 - 4.0j)
```

--------------------------------

### Compute multiple integrals over infinite ranges

Source: https://mpmath.org/doc/current/calculus/integration.html

Example of a 2D integral over infinite ranges.

```python
>>> mp.dps = 15
>>> print(quad(lambda x,y: exp(-x-y), [0, inf], [1, inf]))
0.367879441171442
>>> print(1/e)
0.367879441171442
```

--------------------------------

### Creating mpc instances

Source: https://mpmath.org/doc/current/_sources/basics.rst.txt

Demonstrates construction of complex numbers using the mpc type.

```python
>>> mpc(2,3)
mpc(real='2.0', imag='3.0')
>>> mpc(complex(2,3)).imag
mpf('3.0')
```

--------------------------------

### Using fp context for calculations

Source: https://mpmath.org/doc/current/_sources/contexts.rst.txt

Demonstrates basic usage of the fp context for erfc, matrix operations, and numerical integration.

```python
>>> u = fp.erfc(2.5)
>>> print(u)  # doctest:+SKIP
0.000406952017445
>>> type(u)  # doctest:+SKIP
<type 'float'>
>>> mp.dps = 15
>>> print(mp.erfc(2.5))
0.000406952017444959
>>> fp.matrix([[1,2],[3,4]]) ** 2
matrix(
[['7.0', '10.0'],
 ['15.0', '22.0']])
>>> 
>>> type(_[0,0])  # doctest:+SKIP
<type 'float'>
>>> print(fp.quad(fp.sin, [0, fp.pi]))    # numerical integration
2.0
```

--------------------------------

### Compute 2D integrals

Source: https://mpmath.org/doc/current/calculus/integration.html

Examples of evaluating various 2D integrals to high precision.

```python
>>> mp.dps = 30
>>> f = lambda x, y: (x-1)/((1-x*y)*log(x*y))
>>> quad(f, [0, 1], [0, 1])
0.577215664901532860606512090082
>>> +euler
0.577215664901532860606512090082

>>> f = lambda x, y: 1/sqrt(1+x**2+y**2)
>>> quad(f, [-1, 1], [-1, 1])
3.17343648530607134219175646705
>>> 4*log(2+sqrt(3))-2*pi/3
3.17343648530607134219175646705

>>> f = lambda x, y: 1/(1-x**2 * y**2)
>>> quad(f, [0, 1], [0, 1])
1.23370055013616982735431137498
>>> pi**2 / 8
1.23370055013616982735431137498

>>> quad(lambda x, y: 1/(1-x*y), [0, 1], [0, 1])
1.64493406684822643647241516665
>>> pi**2 / 6
1.64493406684822643647241516665
```

--------------------------------

### GET /altzeta

Source: https://mpmath.org/doc/current/functions/zeta.html

Computes the Dirichlet eta function (alternating zeta function) η(s).

```APIDOC
## GET /altzeta

### Description
Gives the Dirichlet eta function, η(s), defined as the sum of the alternating series η(s)=∑k=0∞(−1)k/(k+1)^s.

### Method
GET

### Parameters
#### Query Parameters
- **s** (number/complex) - Required - The complex argument for the function.
- **rounding** (string) - Optional - Rounding mode for high-precision results.

### Request Example
{ "s": 1 }

### Response
#### Success Response (200)
- **result** (number) - The computed value of the alternating zeta function.

#### Response Example
{ "result": 0.693147180559945 }
```

--------------------------------

### GET hyp1f2

Source: https://mpmath.org/doc/current/functions/hypergeometric.html

Evaluates the hypergeometric function 1F2(a1; b1, b2; z).

```APIDOC
## GET hyp1f2

### Description
Gives the hypergeometric function 1F2(a1; b1, b2; z). This is equivalent to calling hyper([a1], [b1, b2], z).

### Method
GET

### Parameters
#### Path Parameters
- **a1** (number/complex) - Required - First parameter of the hypergeometric function
- **b1** (number/complex) - Required - First denominator parameter
- **b2** (number/complex) - Required - Second denominator parameter
- **z** (number/complex) - Required - The argument of the function
```

--------------------------------

### Create and Configure Matrices

Source: https://mpmath.org/doc/current/_sources/matrices.rst.txt

Initialize matrices using dimensions or nested lists, and manage their dimensions dynamically.

```python
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = False
>>> matrix(2)
matrix(
[['0.0', '0.0'],
 ['0.0', '0.0']])
>>> matrix(2, 3)
matrix(
[['0.0', '0.0', '0.0'],
 ['0.0', '0.0', '0.0']])
```

```python
>>> A = matrix(3, 2)
>>> A
matrix(
[['0.0', '0.0'],
 ['0.0', '0.0'],
 ['0.0', '0.0']])
>>> A.rows
3
>>> A.cols
2
```

```python
>>> A.rows = 2
>>> A
matrix(
[['0.0', '0.0'],
 ['0.0', '0.0']])
```

```python
>>> A = matrix(2)
>>> A[1,1] = 1 + 1j
>>> print(A)
[0.0           0.0]
[0.0  (1.0 + 1.0j)]
```

```python
>>> matrix([[1, 2], [3, 4]])
matrix(
[['1.0', '2.0'],
 ['3.0', '4.0']])
```

--------------------------------

### Incomplete Elliptic Integral E(z,m) Examples

Source: https://mpmath.org/doc/current/functions/elliptic.html

Illustrates basic values, limits, and comparisons with numerical integration for the incomplete elliptic integral of the second kind. Supports complex arguments.

```python
>>> ellipe(0,1)
0.0
>>> ellipe(0,0)
0.0
>>> ellipe(1,0)
1.0
>>> ellipe(2+3j,0)
(2.0 + 3.0j)
>>> ellipe(1,1); sin(1)
0.8414709848078965066525023
0.8414709848078965066525023
>>> ellipe(pi/2, -0.5); ellipe(-0.5)
1.751771275694817862026502
1.751771275694817862026502
>>> ellipe(pi/2, 1); ellipe(-pi/2, 1)
1.0
-1.0
>>> ellipe(1.5, 1)
0.9974949866040544309417234
```

```python
>>> z,m = 0.5, 1.25
>>> ellipe(z,m)
0.4740152182652628394264449
>>> quad(lambda t: sqrt(1-m*sin(t)**2), [0,z])
0.4740152182652628394264449
```

```python
>>> ellipe(3j, 0.5)
(0.0 + 7.551991234890371873502105j)
>>> ellipe(3+4j, 5-6j)
(24.15299022574220502424466 + 75.2503670480325997418156j)
>>> k = 35
>>> z,m = 2+3j, 1.25
>>> ellipe(z+pi*k,m); ellipe(z,m) + 2*k*ellipe(m)
(48.30138799412005235090766 + 17.47255216721987688224357j)
(48.30138799412005235090766 + 17.47255216721987688224357j)
```

```python
>>> z,m = 0.5, 0.25
>>> ellipe(z,m)
0.4950017030164151928870375
>>> sin(z)*appellf1(0.5,0.5,-0.5,1.5,sin(z)**2,m*sin(z)**2)
0.4950017030164151928870376
```

--------------------------------

### GET /mpmath/jacobi

Source: https://mpmath.org/doc/current/_sources/functions/orthogonal.rst.txt

Calculates the Jacobi polynomial of degree n with parameters a and b at z.

```APIDOC
## GET /mpmath/jacobi

### Description
Computes the Jacobi polynomial P_n^(a,b)(z).

### Method
GET

### Parameters
#### Query Parameters
- **n** (int) - Required - The degree of the polynomial
- **a** (float) - Required - Parameter a
- **b** (float) - Required - Parameter b
- **z** (float/complex) - Required - The point at which to evaluate
```

--------------------------------

### Example: hyper2d() for Horn functions G1, G2, G3

Source: https://mpmath.org/doc/current/functions/hypergeometric.html

Demonstrates the computation of Horn functions G1, G2, and G3 using hyper2d(). Requires mpmath to be imported and sets precision to 10 decimal places.

```python
>>> from mpmath import *
>>> mp.dps = 10; mp.pretty = True
>>> x, y = 0.0625, 0.125
>>> a1,a2,b1,b2,c1,c2,d = 1.1,-1.2,-1.3,-1.4,1.5,-1.6,1.7
>>> hyper2d({'m+n':a1,'n-m':b1,'m-n':b2},{},x,y)  # G1
1.139090746
>>> nsum(lambda m,n: rf(a1,m+n)*rf(b1,n-m)*rf(b2,m-n)*\
...     x**m*y**n/fac(m)/fac(n), [0,inf], [0,inf])
1.139090746
>>> hyper2d({'m':a1,'n':a2,'n-m':b1,'m-n':b2},{},x,y)  # G2
0.9503682696
>>> nsum(lambda m,n: rf(a1,m)*rf(a2,n)*rf(b1,n-m)*rf(b2,m-n)*\
...     x**m*y**n/fac(m)/fac(n), [0,inf], [0,inf])
0.9503682696
>>> hyper2d({'2n-m':a1,'2m-n':a2},{},x,y)  # G3
1.029372029
>>> nsum(lambda m,n: rf(a1,2*n-m)*rf(a2,2*m-n)*\
...     x**m*y**n/fac(m)/fac(n), [0,inf], [0,inf])
1.029372029
```

--------------------------------

### Compute Product from Negative Infinity

Source: https://mpmath.org/doc/current/calculus/sums_limits.html

Calculate a product where the interval starts from negative infinity.

```python
nprod(lambda k: 1-k**(-3), [-inf,-2])
```

```python
cosh(pi*sqrt(3)/2)/(3*pi)
```

--------------------------------

### Evaluate Irregular Coulomb Wave Function

Source: https://mpmath.org/doc/current/functions/bessel.html

Examples of evaluating the irregular Coulomb wave function.

```python
>>> from mpmath import *
>>> mp.dps = 25; mp.pretty = True
>>> coulombg(-2, 1.5, 3.5)
1.380011900612186346255524
>>> coulombg(2, 1.5, 3.5)
1.919153700722748795245926
>>> coulombg(-2, 1.5, '1e-10')
201126715824.7329115106793
>>> coulombg(-2, 1.5, 1000)
0.1802071520691149410425512
>>> coulombg(-2, 1.5, 10**10)
0.652103020061678070929794
```

--------------------------------

### Importing mpmath

Source: https://mpmath.org/doc/current/_sources/basics.rst.txt

Demonstrates different ways to import mpmath functionality to avoid namespace collisions.

```python
>>> from mpmath import *
```

```python
from mpmath import sin, cos
sin(1), cos(1)

import mpmath
mpmath.sin(1), mpmath.cos(1)

from mpmath import mp    # mp context object -- to be explained
mp.sin(1), mp.cos(1)
```

--------------------------------

### Double factorial usage

Source: https://mpmath.org/doc/current/functions/gamma.html

Examples of double factorial calculations, asymptotic behavior, and recurrence relations.

```python
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> nprint([fac2(n) for n in range(10)])
[1.0, 1.0, 2.0, 3.0, 8.0, 15.0, 48.0, 105.0, 384.0, 945.0]
```

```python
>>> x = mpf(10000)
>>> fac2(x)
5.97272691416282e+17830
>>> sqrt(pi)*x**((x+1)/2)*exp(-x/2)
5.97262736954392e+17830
```

```python
>>> fac2(-1), fac2(-3), fac2(-5), fac2(-7)
(1.0, -1.0, 0.333333333333333, -0.0666666666666667)
>>> fac2(-2)
Traceback (most recent call last):
  ...
ValueError: gamma function pole
```

```python
>>> fac2(pi+2j)
(-1.3697207890154e-12 + 3.93665300979176e-12j)
>>> (pi+2j)*fac2(pi-2+2j)
(-1.3697207890154e-12 + 3.93665300979176e-12j)
```

```python
>>> fac(fac(20))
5.13805976125208e+43675043585825292774
>>> fac2(20)
3715891200.0
```

```python
>>> nsum(lambda k: 1/fac2(k), [0, inf])
3.05940740534258
>>> sqrt(e)*(1+sqrt(pi/2)*erf(sqrt(2)/2))
3.05940740534258
>>> nsum(lambda k: 2**k/fac2(2*k-1), [1, inf])
4.06015693855741
>>> e * erf(1) * sqrt(pi)
4.06015693855741
```

```python
>>> nsum(lambda k: (-1)**k*(fac2(2*k-1)/fac2(2*k))**3, [0,inf])
0.90917279454693
>>> (gamma('9/8')/gamma('5/4')/gamma('7/8'))**2
0.90917279454693
```

--------------------------------

### Creating mpf real numbers with mp and fp contexts

Source: https://mpmath.org/doc/current/contexts.html

Shows how to create arbitrary-precision real numbers using the `mpf` method from both the `mp` and `fp` contexts.

```python
>>> from mpmath import mp, fp
>>> mp.mpf(3)
mpf('3.0')
```

```python
>>> fp.mpf(3)
3.0
```

--------------------------------

### Initialize matrices with dimensions

Source: https://mpmath.org/doc/current/matrices.html

Create empty matrices by specifying row and column counts.

```python
>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = False
>>> matrix(2)
matrix(
[['0.0', '0.0'],
 ['0.0', '0.0']])
>>> matrix(2, 3)
matrix(
[['0.0', '0.0', '0.0'],
 ['0.0', '0.0', '0.0']])
```

--------------------------------

### Example: hyper2d() for Horn functions H1, H2, H3

Source: https://mpmath.org/doc/current/functions/hypergeometric.html

Demonstrates the computation of Horn functions H1, H2, and H3 using hyper2d(). Requires mpmath to be imported and sets precision to 10 decimal places.

```python
>>> hyper2d({'m-n':a1,'m+n':b1,'n':c1},{'m':d},x,y)  # H1
-1.605331256
>>> nsum(lambda m,n: rf(a1,m-n)*rf(b1,m+n)*rf(c1,n)/rf(d,m)*\
...     x**m*y**n/fac(m)/fac(n), [0,inf], [0,inf])
-1.605331256
>>> hyper2d({'m-n':a1,'m':b1,'n':[c1,c2]},{'m':d},x,y)  # H2
-2.35405404
>>> nsum(lambda m,n: rf(a1,m-n)*rf(b1,m)*rf(c1,n)*rf(c2,n)/rf(d,m)*\
...     x**m*y**n/fac(m)/fac(n), [0,inf], [0,inf])
-2.35405404
>>> hyper2d({'2m+n':a1,'n':b1},{'m+n':c1},x,y)  # H3
0.974479074
>>> nsum(lambda m,n: rf(a1,2*m+n)*rf(b1,n)/rf(c1,m+n)*\
...     x**m*y**n/fac(m)/fac(n), [0,inf], [0,inf])
0.974479074
```

--------------------------------

### mpmath.levin()

Source: https://mpmath.org/doc/current/calculus/sums_limits.html

Initializes a Levin or Sidi-S sequence transformation object.

```APIDOC
## mpmath.levin(method='levin', variant='u')

### Description
Initializes an object to perform Levin's nonlinear sequence transformation for convergence acceleration and summation of divergent series.

### Parameters
#### Query Parameters
- **method** (string) - Optional - The transformation method to use: 'levin' or 'sidi'. Defaults to 'levin'.
- **variant** (string) - Optional - The weight variant to use: 'u', 't', or 'v'. Defaults to 'u'.

### Methods
- **update(terms)**: Updates the transformation with a list of individual terms (ak).
- **update_psum(sums)**: Updates the transformation with a list of partial sums (sk).
- **step(ak)**: Updates the transformation with a single term (ak).
- **step_psum(sk)**: Updates the transformation with a single partial sum (sk).

### Response
Returns an object containing the current estimate (v) and an error estimate (e).
```
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