scipy tutorial - Baustatik-Info-Server

SciPy Reference Guide, Release 0.8.dev
0.56714329040978387299996866221035555 >>> w*exp(w) 1.0
Any branch gives a valid inverse:
>>> w = lambertw(1, k=3)
>>> w
(-2.8535817554090378072068187234910812 + 17.113535539412145912607826671159289j) >>> w*exp(w)
(1.0 + 3.5075477124212226194278700785075126e-36j)
Applications to equation-solving
The Lambert W function may be used to solve various kinds of equations, such as finding the value of the infinite
power tower z^{z^{z^{ldots}}}:
>>> def tower(z, n):
... if n == 0: ... return z ... return z ** tower(z, n-1) ... >>> tower(0.5, 100) 0.641185744504986 >>> -lambertw(log(0.5))/log(0.5)
0.6411857445049859844862004821148236665628209571911
Properties
The Lambert W function grows roughly like the natural logarithm for large arguments:
>>> lambertw(1000)
5.2496028524016 >>> log(1000) 6.90775527898214 >>> lambertw(10**100) 224.843106445119 >>>
log(10**100) 230.258509299405
The principal branch of the Lambert W function has a rational Taylor series expansion around z = 0:
>>> nprint(taylor(lambertw, 0, 6), 10)
[0.0, 1.0, -1.0, 1.5, -2.666666667, 5.208333333, -10.8]
Some special values and limits are:
>>> lambertw(0)
0.0 >>> lambertw(1) 0.567143290409784 >>> lambertw(e) 1.0 >>> lambertw(inf) +inf >>> lambertw(0, k=-1)
-inf >>> lambertw(0, k=3) -inf >>> lambertw(inf, k=3) (+inf + 18.8495559215388j)
The k = 0 and k = -1 branches join at z = -1/e where W(z) = -1 for both branches. Since -1/e can only be
represented approximately with mpmath numbers, evaluating the Lambert W function at this point only gives -1
approximately:
>>> lambertw(-1/e, 0)
-0.999999999999837133022867 >>> lambertw(-1/e, -1) -1.00000000000016286697718
If -1/e happens to round in the negative direction, there might be a small imaginary part:
>>> lambertw(-1/e)
(-1.0 + 8.22007971511612e-9j)
zeta
y=zeta(x,q) returns the Riemann zeta function of two arguments: sum((k+q)**(-x),k=0..inf)
446 Chapter 3. Reference