Derive The Asymptotic Expansion Of The Gauss Error Function
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that occurs in probability, statistics, and partial differential equations describing diffusion. It is defined as:[1][2] erf ( x ) = 1 π ∫ − x x e − t 2 d t = 2 π ∫ derivative of error function 0 x e − t 2 d t . {\displaystyle {\begin − 6\operatorname − 5
Erf Function Calculator
(x)&={\frac − 4{\sqrt {\pi }}}\int _{-x}^ − 3e^{-t^ − 2}\,\mathrm − 1 t\\&={\frac − 0{\sqrt {\pi }}}\int _ 9^ error function table 8e^{-t^ 7}\,\mathrm 6 t.\end 5}} The complementary error function, denoted erfc, is defined as erfc ( x ) = 1 − erf ( x ) = 2 π ∫ x ∞
Inverse Error Function
e − t 2 d t = e − x 2 erfcx ( x ) , {\displaystyle {\begin 2\operatorname 1 (x)&=1-\operatorname 0 (x)\\&={\frac Φ 9{\sqrt {\pi }}}\int _ Φ 8^{\infty }e^{-t^ Φ 7}\,\mathrm Φ 6 t\\&=e^{-x^ Φ 5}\operatorname Φ 4 (x),\end Φ 3}} which also defines erfcx, the scaled complementary error function[3] (which can be used instead of erfc to avoid arithmetic underflow[3][4]). Another form of erfc erf(inf) ( x ) {\displaystyle \operatorname 2 (x)} for non-negative x {\displaystyle x} is known as Craig's formula:[5] erfc ( x | x ≥ 0 ) = 2 π ∫ 0 π / 2 exp ( − x 2 sin 2 θ ) d θ . {\displaystyle \operatorname 0 (x|x\geq 0)={\frac Φ 9{\pi }}\int _ Φ 8^{\pi /2}\exp \left(-{\frac Φ 7}{\sin ^ Φ 6\theta }}\right)d\theta \,.} The imaginary error function, denoted erfi, is defined as erfi ( x ) = − i erf ( i x ) = 2 π ∫ 0 x e t 2 d t = 2 π e x 2 D ( x ) , {\displaystyle {\begin Φ 0\operatorname − 9 (x)&=-i\operatorname − 8 (ix)\\&={\frac − 7{\sqrt {\pi }}}\int _ − 6^ − 5e^ − 4}\,\mathrm − 3 t\\&={\frac − 2{\sqrt {\pi }}}e^ − 1}D(x),\end − 0}} where D(x) is the Dawson function (which can be used instead of erfi to avoid arithmetic overflow[3]). Despite the name "imaginary error function", erfi ( x ) {\displaystyle \operatorname 8 (x)} is real when x is real. When the error function is evaluated for arbitrary complex arguments z, the resulting complex error function is usually discussed in scaled form as the Faddeeva fu
van GoogleInloggenVerborgen veldenBoekenbooks.google.nl - Mathematical Methods for Physicists, Third Edition provides an advanced undergraduate and beginning graduate study in physical science, focusing on https://books.google.com/books?id=X9U3BQAAQBAJ&pg=PA344&lpg=PA344&dq=derive+the+asymptotic+expansion+of+the+gauss+error+function&source=bl&ots=YirUGgVbFE&sig=vMSKa-XBu9Y9_TtC4ZGDtwDdHMY&hl=en&sa=X&ved=0ahUKEwjuiN-Ov8DPA the mathematics of theoretical physics. This edition includes sections on the non-Cartesian tensors, dispersion theory, first-order differential equations,...https://books.google.nl/books/about/Mathematical_Methods_for_Physicists.html?hl=nl&id=X9U3BQAAQBAJ&utm_source=gb-gplus-shareMathematical Methods for PhysicistsMijn bibliotheekHelpGeavanceerd zoeken naar boekeneBoek kopen - € 60,09Dit boek in gedrukte vorm bestellenAccess Online via ElsevierBol.comProxis.nlselexyz.nlVan StockumZoeken in een bibliotheekAlle verkopers»Mathematical Methods for PhysicistsGeorge B. ArfkenAcademic Press, 22 error function okt. 2013 - 1008 pagina's 0 Recensieshttps://books.google.nl/books/about/Mathematical_Methods_for_Physicists.html?hl=nl&id=X9U3BQAAQBAJMathematical Methods for Physicists, Third Edition provides an advanced undergraduate and beginning graduate study in physical science, focusing on the mathematics of theoretical physics. This edition includes sections on the non-Cartesian tensors, dispersion theory, first-order differential equations, numerical application of Chebyshev polynomials, the derive the asymptotic fast Fourier transform, and transfer functions. Many of the physical examples provided in this book, which are used to illustrate the applications of mathematics, are taken from the fields of electromagnetic theory and quantum mechanics. The Hermitian operators, Hilbert space, and concept of completeness are also deliberated. This book is beneficial to students studying graduate level physics, particularly theoretical physics. Voorbeeld weergeven » Wat mensen zeggen-Een recensie schrijvenWe hebben geen recensies gevonden op de gebruikelijke plaatsen.Geselecteerde pagina'sPagina 25Pagina 15Pagina 38Pagina 9Pagina 4InhoudsopgaveCHAPTER 1 VECTOR ANALYSIS1 CHAPTER 2 COORDINATE SYSTEMS85 CHAPTER 3 TENSOR ANALYSIS118 CHAPTER 4 DETERMINANTS MATRICES AND GROUP THEORY168 CHAPTER 5 INFINITE SERIES277 CHAPTER 6 FUNCTIONS OF A COMPLEX VARIABLE I352 CHAPTER 7 FUNCTIONS OF A COMPLEX VARIABLE II396 CHAPTER 8 DIFFERENTIAL EQUATIONS437 CHAPTER 12 LEGENDRE FUNCTIONS637 CHAPTER 13 SPECIAL FUNCTIONS712 CHAPTER 14 FOURIER SERIES760 CHAPTER 15 INTEGRAL TRANSFORMS794 CHAPTER 16 INTEGRA LEQUATI