scipy.special.hyp0f1#
- scipy.special.hyp0f1(v, z, out=None) = <ufunc 'hyp0f1'>#
Confluent hypergeometric limit function 0F1.
- Parameters
- varray_like
Real-valued parameter
- zarray_like
Real- or complex-valued argument
- outndarray, optional
Optional output array for the function results
- Returns
- scalar or ndarray
The confluent hypergeometric limit function
Notes
This function is defined as:
\[_0F_1(v, z) = \sum_{k=0}^{\infty}\frac{z^k}{(v)_k k!}.\]It’s also the limit as \(q \to \infty\) of \(_1F_1(q; v; z/q)\), and satisfies the differential equation \(f''(z) + vf'(z) = f(z)\). See [1] for more information.
References
- 1
Wolfram MathWorld, “Confluent Hypergeometric Limit Function”, http://mathworld.wolfram.com/ConfluentHypergeometricLimitFunction.html
Examples
>>> import scipy.special as sc
It is one when z is zero.
>>> sc.hyp0f1(1, 0) 1.0
It is the limit of the confluent hypergeometric function as q goes to infinity.
>>> q = np.array([1, 10, 100, 1000]) >>> v = 1 >>> z = 1 >>> sc.hyp1f1(q, v, z / q) array([2.71828183, 2.31481985, 2.28303778, 2.27992985]) >>> sc.hyp0f1(v, z) 2.2795853023360673
It is related to Bessel functions.
>>> n = 1 >>> x = np.linspace(0, 1, 5) >>> sc.jv(n, x) array([0. , 0.12402598, 0.24226846, 0.3492436 , 0.44005059]) >>> (0.5 * x)**n / sc.factorial(n) * sc.hyp0f1(n + 1, -0.25 * x**2) array([0. , 0.12402598, 0.24226846, 0.3492436 , 0.44005059])