In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the...
12 KB (1,960 words) - 21:23, 15 May 2024
In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing limiting behavior. As an illustration, suppose that...
17 KB (2,763 words) - 08:12, 22 May 2024
In mathematics, the method of matched asymptotic expansions is a common approach to finding an accurate approximation to the solution to an equation,...
20 KB (3,217 words) - 14:58, 17 April 2024
Error function (section Asymptotic expansion)
this asymptotic expansion are needed to obtain a good approximation of erfc x (while for not too large values of x, the above Taylor expansion at 0 provides...
48 KB (7,352 words) - 19:10, 30 May 2024
and only large values of x are employed. This expansion follows directly from the asymptotic expansion for the exponential integral. This implies e.g...
6 KB (1,104 words) - 21:51, 7 August 2024
Polygamma function (section Asymptotic expansion)
In mathematics, the polygamma function of order m is a meromorphic function on the complex numbers C {\displaystyle \mathbb {C} } defined as the (m + 1)th...
12 KB (2,364 words) - 05:12, 9 August 2024
Lerch zeta function (section Asymptotic expansion)
\mathbb {C} } and z ∈ Ω a {\displaystyle z\in \Omega _{a}} , an asymptotic expansion of Φ ( z , s , a ) {\displaystyle \Phi (z,s,a)} for large a {\displaystyle...
16 KB (3,490 words) - 09:53, 24 April 2024
Euler's constant (section Asymptotic expansions)
the digamma function A product formula for the gamma function The asymptotic expansion of the gamma function for small arguments. An inequality for Euler's...
51 KB (7,581 words) - 20:07, 30 July 2024
Hermite polynomials (section Asymptotic expansion)
_{-\infty }^{\infty }(x+iy)^{n}e^{-{\frac {y^{2}}{2}}}\,dy.} Asymptotically, as n → ∞, the expansion e − x 2 2 ⋅ H n ( x ) ∼ 2 n π Γ ( n + 1 2 ) cos ( x 2...
57 KB (10,024 words) - 18:31, 4 August 2024
Digamma function (section Asymptotic expansion)
integral representation can be manipulated to give the start of the asymptotic expansion of ψ {\displaystyle \psi } . ψ ( z ) = log z − 1 2 z − ∫ 0 ∞ (...
35 KB (7,082 words) - 05:21, 9 August 2024