mpmath.md

mpmath

Version: 1.4.1 | Type: Stock (pure Python) | Status: Fully working

Arbitrary-precision floating-point math. Used by SymPy internally.

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Quick Start

from mpmath import mp, mpf, pi, e, sqrt, sin, cos, exp, log, gamma, zeta, quad, nstr

mp.dps = 50  # 50 decimal places
print(f"pi = {pi}")
print(f"e  = {e}")

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Precision Control

Setting	Description
mp.dps = n	Set decimal places (digits of precision)
mp.prec = n	Set binary precision (bits)
mpf(x)	Create multi-precision float
mpc(re, im)	Create multi-precision complex
nstr(x, n)	Format to n significant digits
mpf('0.1')	Exact decimal input (avoids float rounding)

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Constants

Constant	Description
pi	3.14159...
e	2.71828... (Euler's number)
euler	0.57721... (Euler-Mascheroni)
catalan	0.91596... (Catalan's constant)
phi	1.61803... (golden ratio)
khinchin	2.68545... (Khinchin's constant)
glaisher	1.28242... (Glaisher-Kinkelin)
apery	1.20205... (Apery's constant = zeta(3))
degree	pi/180
inf	Positive infinity
nan	Not a number
j	Imaginary unit

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Arithmetic & Power Functions

Function	Description
sqrt(x)	Square root
cbrt(x)	Cube root
root(x, n)	N-th root
power(x, y)	x^y
exp(x)	Exponential
expm1(x)	exp(x) - 1 (precise near 0)
log(x) / ln(x)	Natural logarithm
log10(x)	Base-10 logarithm
log(x, b)	Logarithm base b
fabs(x)	Absolute value
sign(x)	Sign function
floor(x) / ceil(x)	Floor / ceiling
nint(x)	Nearest integer
frac(x)	Fractional part
fmod(x, y)	Floating-point modulo
ldexp(x, n)	x * 2^n
frexp(x)	Decompose to (m, e) where x = m * 2^e

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Trigonometric & Hyperbolic

Function	Description
sin, cos, tan	Trigonometric
cot, sec, csc	Reciprocal trig
asin, acos, atan	Inverse trig
atan2(y, x)	Two-argument arctangent
sinh, cosh, tanh	Hyperbolic
coth, sech, csch	Reciprocal hyperbolic
asinh, acosh, atanh	Inverse hyperbolic
sinpi(x) / cospi(x)	sin(pix), cos(pix) (exact at integers)
sincpi(x)	sin(pix)/(pix)
degrees(x) / radians(x)	Angle conversion

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Special Functions

Function	Description
gamma(z)	Gamma function
rgamma(z)	1/gamma(z)
loggamma(z)	Log-gamma
factorial(n)	n!
fac2(n)	Double factorial n!!
rf(x, n)	Rising factorial (Pochhammer)
ff(x, n)	Falling factorial
binomial(n, k)	Binomial coefficient
beta(a, b)	Beta function
betainc(a, b, x1, x2)	Incomplete beta
psi(n, z)	Polygamma (digamma when n=0)
digamma(z)	Digamma function
harmonic(n)	Harmonic number
bernoulli(n)	Bernoulli numbers
euler(n)	Euler numbers

Zeta & L-Functions

Function	Description
zeta(s)	Riemann zeta function
altzeta(s)	Dirichlet eta function
polylog(s, z)	Polylogarithm
lerchphi(z, s, a)	Lerch transcendent
dirichlet(s, chi)	Dirichlet L-function
stieltjes(n)	Stieltjes constants

Error & Exponential Integrals

Function	Description
erf(z)	Error function
erfc(z)	Complementary error function
erfi(z)	Imaginary error function
erfinv(z)	Inverse error function
ei(x)	Exponential integral Ei
li(x)	Logarithmic integral
si(x) / ci(x)	Sine/cosine integrals
shi(x) / chi(x)	Hyperbolic sine/cosine integrals

Bessel & Airy

Function	Description
besselj(v, z)	Bessel J (first kind)
bessely(v, z)	Bessel Y (second kind)
besseli(v, z)	Modified Bessel I
besselk(v, z)	Modified Bessel K
hankel1(v, z) / hankel2(v, z)	Hankel functions
airyai(z) / airybi(z)	Airy functions
airyaizero(n) / airybizero(n)	Zeros of Airy functions

Hypergeometric

Function	Description
hyp0f1(b, z)	Confluent hypergeometric limit
hyp1f1(a, b, z)	Kummer confluent hypergeometric
hyp2f1(a, b, c, z)	Gauss hypergeometric
hyper(a_s, b_s, z)	Generalized hypergeometric pFq
meijerg(a, b, z)	Meijer G-function

Elliptic Functions

Function	Description
ellipk(m)	Complete elliptic integral K
ellipe(m)	Complete elliptic integral E
ellipf(phi, m)	Incomplete elliptic F
ellippi(n, m)	Complete elliptic Pi
jtheta(n, z, q)	Jacobi theta functions (n=1,2,3,4)
kleinj(tau)	Klein j-invariant

Orthogonal Polynomials

legendre(n, x), chebyt(n, x), chebyu(n, x), hermite(n, x), laguerre(n, x, m), gegenbauer(n, a, x), jacobi(n, a, b, x), spherharm(l, m, theta, phi)

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Numerical Calculus

Function	Description
quad(f, [a, b])	Numerical integration (adaptive quadrature)
quadgl(f, [a, b])	Gauss-Legendre quadrature
quadts(f, [a, b])	Tanh-sinh quadrature
quadosc(f, [a, inf], omega)	Oscillatory integral
diff(f, x, n)	N-th numerical derivative
diffs(f, x, n)	All derivatives up to order n
taylor(f, x, n)	Taylor coefficients
pade(coeffs, L, M)	Pade approximant
nsum(f, [a, b])	Numerical summation
nprod(f, [a, b])	Numerical product
limit(f, x)	Richardson extrapolation limit
richardson(f, n, N)	Richardson extrapolation

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Linear Algebra

Function	Description
matrix(rows)	Create matrix
eye(n)	Identity matrix
zeros(m, n)	Zero matrix
ones(m, n)	Ones matrix
diag(entries)	Diagonal matrix
lu_solve(A, b)	Solve via LU
qr_solve(A, b)	Solve via QR
cholesky_solve(A, b)	Solve via Cholesky
det(A)	Determinant
inverse(A) / A**-1	Matrix inverse
norm(A, p)	Matrix/vector norm
mnorm(A, p)	Matrix norm
eig(A)	Eigenvalues
eigsy(A)	Eigenvalues (symmetric)
svd(A)	Singular value decomposition
svd_r(A)	Economy SVD
qr(A)	QR decomposition
lu(A)	LU decomposition
cholesky(A)	Cholesky decomposition
hessenberg(A)	Hessenberg reduction
schur(A)	Schur decomposition
expm(A)	Matrix exponential
logm(A)	Matrix logarithm
sqrtm(A)	Matrix square root
powm(A, n)	Matrix power

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Root Finding & Optimization

Function	Description
findroot(f, x0)	Find root using Newton or secant method
findroot(f, [a, b])	Find root in interval
polyroots(coeffs)	Polynomial roots (arbitrary precision)
polyval(coeffs, x)	Evaluate polynomial

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Number Theory

Function	Description
isprime(n)	Primality test
primepi(n)	Prime counting function
primepi2(n)	Prime counting with error bound
bell(n)	Bell numbers
stirling1(n, k)	Stirling numbers (first kind)
stirling2(n, k)	Stirling numbers (second kind)
moebius(n)	Mobius function

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Series & Transforms

Function	Description
taylor(f, x, n)	Taylor coefficients
chebyfit(f, [a, b], n)	Chebyshev approximation
fourier(f, [a, b], n)	Fourier coefficients
nsum(f, [a, inf])	Numerical infinite sum
nprod(f, [a, inf])	Numerical infinite product