LAMBERTOVA FUNKCIJA W

Pangrčič, Igor

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Title:	LAMBERTOVA FUNKCIJA W
Authors:	ID Pangrčič, Igor (Author) ID Milutinović, Uroš (Mentor) More about this mentor...
Files:	UNI_Pangrcic_Igor_2012.pdf (2,21 MB) MD5: 5ED0A3901C614BC66183F08717158E90 PID: 20.500.12556/dkum/cd5e99d9-8420-4264-809a-eed6dfebf1e3
Language:	Slovenian
Work type:	Undergraduate thesis
Organization:	FNM - Faculty of Natural Sciences and Mathematics
Abstract:	Lambertova funkcija W je inverzna funkcija funkcije f(w)=we^w, kjer je ew naravna eksponentna funkcija in w kompleksno število. Imenuje se po Johannu Heinrichu Lambertu. Tu je funkcija označena z W. To oznako sta prva uporabila Pólya in Szegő leta 1925. Za vsako kompleksno število z velja: z=W(z)e^(W(z)). Ker funkcija f v (−∞, 0) ni injektivna, zavzame funkcija W v [−1/e, 0) več vrednosti. Če se omejimo na realne argumente x ≥ −1/e in zahtevamo w ≥ −1, je na ta način določena funkcija W0(x) z enoličnimi vrednostmi. Velja W0(0) = 0 in W0(−1/e) = −1. Lambertove funkcije W ne moremo izraziti s členi elementarnih funkcij. Funkcija je uporabna v kombinatoriki, na primer pri preštevanju dreves. Z njo lahko rešimo različne enačbe, ki vsebujejo eksponente. Pojavlja se pri reševanju časovno zakasnelih diferencialnih enačb, kot je na primer y^' (t)=ay(t-1).
Keywords:	Večlična funkcija, enolična funkcija, eksponentna in inverzna funkcija, potenčne vrste, konvergenčni polmer, L'Hospitalovo pravilo, kvocientni kriterij, Cayleyev izrek, ukoreninjeno drevo.
Place of publishing:	Maribor
Publisher:	[I. Pangrčič]
Year of publishing:	2012
PID:	20.500.12556/DKUM-22031
UDC:	51(043.2)
COBISS.SI-ID:	18938888
NUK URN:	URN:SI:UM:DK:PIVABDMW
Publication date in DKUM:	27.02.2012
Views:	3568
Downloads:	177
Metadata:
Categories:	FNM
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Secondary language

Language:	English
Title:	THE LAMBERT W FUNCTION
Abstract:	The Lambert W function is set-valued function, namely the branches of the inverse relation of the function f(w)=we^w where ew is the exponential function and w is any complex number. The function is named after Johann Heinrich Lambert and is denote W which was first used by Pólya and Szegő 1925. In other words, the defining equation for W(z) is z=W(z)e^(W(z)) for any complex number z. Since the function ƒ is not injective, the relation W is multivalued (except at 0). If we restrict attention to real-valued W then the relation is defined only for x ≥ −1/e, and is double-valued on (−1/e, 0); the additional constraint W ≥ −1 defines a single-valued function W0(x). We have W0(0) = 0 and W0(−1/e) = −1. The Lambert W relation cannot be expressed in terms of elementary functions. It is useful in combinatorics, for instance in the enumeration of trees. It can be used to solve various equations involving exponentials and also occurs in the solution of delay differential equations, such as y^' (t)=ay(t-1).
Keywords:	Multivalued, unique, exponential and inverse function, power series, radius of convergence, L'Hospital's rule, quotient criterion, Cayley theorem, rooted tree.

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