Lambert W fonksiyonu

Matematikte, Lambert W fonksiyonu', aynı zamanda Omega fonksiyonu veya çarpım logaritması olarak da bilinen bir fonksiyon kümesidir. f(w) = wew fonksiyonunda ew üstel fonksiyon ve w herhangi bir karmaşık sayı olmak üzere, bu fonksiyonun tersinin şubelerini ifade eder.

$z=W(z)e^{W(z)}$

$Y=Ae^{A}\;\Longleftrightarrow \;A=W(Y)$

$z(1+W){\frac {{\rm {d}}W}{{\rm {d}}z}}=W\quad {\text{for }}z\neq -1/e.$

${\frac {{\rm {d}}W}{{\rm {d}}z}}={\frac {W(z)}{z(1+W(z))}}\quad {\text{for }}z\not \in \{0,-1/e\}.$

$\left.{\frac {{\rm {d}}W}{{\rm {d}}z}}\right|_{z=0}=1.$

W(x) Fonksiyonun integrali şu şekildedir.

$\int W(a)\,{\rm {d}}a=a\left(W(a)-1+{\frac {1}{W(a)}}\right)+C.$

Lambert W Fonksiyonun Serisi:

$W(a)=\sum _{n=1}^{\infty }{\frac {(-n)^{n-1}}{n!}}\ a^{n}=a-a^{2}+{\frac {3}{2}}a^{3}-{\frac {8}{3}}a^{4}+{\frac {125}{24}}a^{5}-\cdots$ .

Doğal logaritma tabanı e w türünden özelliği: $e^{W(a)}=\left({\frac {a}{W(a)}}\right)$ İntegrali ise:

$\int e^{W(a)}\,{\rm {d}}a=a^{2}(1+2W(a)){\frac {1}{4W(a)^{2}}}+C.$

Bazı Değerler

$W\left(-{\frac {\pi }{2}}\right)={\frac {\pi }{2}}{\rm {i}}$

$W\left(-{\frac {\ln a}{a}}\right)=-\ln a\quad \left({\frac {1}{e}}\leq a\leq e\right)$

$W\left(-{\frac {1}{e}}\right)=-1$

$W\left(0\right)=0\,$

$W\left(1\right)=\Omega ={\frac {1}{\displaystyle \int _{-\infty }^{\infty }{\frac {{\rm {d}}x}{(e^{x}-x)^{2}+\pi ^{2}}}}}-1\approx 0.56714329\dots \,$ (Omega Sabiti)

$W\left(e\right)=1\,$

$W\left(-1\right)\approx -0.31813-1.33723{\rm {i}}\,$

$W'\left(0\right)=1\,$

$W\left(2\right)=0.852605502013725...\,$

${\displaystyle }$

Kaynakça

• Corless, R.; Gonnet, G.; Hare, D.; Jeffrey, D.; Knuth, Donald (1996). "On the Lambert W function" (PDF). Advances in Computational Mathematics. Cilt 5. Berlin, New York: Springer-Verlag. ss. 329–359. doi:10.1007/BF02124750. ISSN 1019-7168Şablon:İnconsistent citations 
• Scott, T.C.; Mann, R.B.; Martinez Ii, Roberto E. (2006). "General Relativity and Quantum Mechanics: Towards a Generalization of the Lambert W Function". AAECC (Applicable Algebra in Engineering, Communication and Computing). 17 (1). ss. 41–47. arXiv:math-ph/0607011 $2. doi:10.1007/s00200-006-0196-1. 
• Chapeau-Blondeau, F. and Monir, A: "Evaluation of the Lambert W Function and Application to Generation of Generalized Gaussian Noise With Exponent 1/2", IEEE Trans. Signal Processing, 50(9), 2002
• Francis et al. "Quantitative General Theory for Periodic Breathing" Circulation 102 (18): 2214. (2000). Use of Lambert function to solve delay-differential dynamics in human disease.
• Şablon:Dlmf
• Veberic, D., "Having Fun with Lambert W(x) Function" arXiv:1003.1628 (2010). C++ implementation using Halley's and Fritsch's iteration.
• National Institute of Science and Technology Digital Library - Lambert W
• MathWorld - Lambert W-Function
• Computing the Lambert W function
• Corless et al. Notes about Lambert W research
• Extreme Mathematics. Monographs on the Lambert W function, its numerical approximation and generalizations for W-like inverses of transcendental forms with repeated exponential towers.
• GPL C++ implementation with Halley's and Fritsch's iteration.