Lambert W Fonksiyonu

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

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

$Y = A e ^ A \; \Longleftrightarrow \; A = W(Y)$

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

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

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

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:

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

Bazı Değerler

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

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

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

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

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

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

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

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

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

Kaynakça

• Corless, R.; Gonnet, G.; Hare, D.; Jeffrey, D.; Knuth, Donald (1996). "On the Lambert W function". Advances in Computational Mathematics (Berlin, New York: Springer-Verlag) 5: 329–359. doi:10.1007/BF02124750. ISSN 1019-7168. http://www.apmaths.uwo.ca/~djeffrey/Offprints/W-adv-cm.pdfŞablon:İnconsistent citations 
• 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.