Annotations to Abramowitz & Stegun

Transcription

1 Aotatios to Abraowitz & Stegu By Lias Vepstas 4 Jauary 4 corrected Dec 4, Dec The followig is a copediu of additios ad argi otes to the Hadbook of Matheatical Fuctios by Abraowitz & Stegu Dover 97 editio, culled fro persoal aotatios I have ade to that referece over the years. I have foud these forulas useful ad hady to have aroud. May are trivial restateets of what ca already be foud i the book, ad a few are deeper, o-trivial relatioships. Most of these are ot atheatically sigificat, but are useful if oe is just searchig for a itegral or soe such: ideed, this is what it eas to be a referece. They are put dow here to be of soe utility to the Iteret couity. It would be ice if future editios/revisios of the A&S referece were possible, ad were to iclude such updates. Sources & attributio: I derived all of these. I did ot copy ay of these fro soe other book/referece, except as oted. I ve tripped over these while solvig a large variety of other copletely urelated, but quite iterestig probles. These additioal forulas are ordered accordig to the relevat chapter/paragraph of that book. Parethetical coets justify the eed for the iclusio of the forula, but are ot eat to be added to the referece. Without further ado: 3. Eleetary Aalytical Methods a x α = α + k k x k hady restateet of i a o-obvious for 3.7.-a argx + iy = arctax/y

2 Just because arcta coes i a later chapter is o excuse to oit this very useful forula 4. Eleetary Trascedetal Fuctios 4..5-a Discotiuity across the Brach Cut l x + iε l x iε = πi + Oε for real x >, ad sall, real ε. This follows obviously fro 4..5 but is hady esp. for ovice b liε l iε = πi + Oε for sall, real ε. No-ituitive stateet about the liit o the iagiary axis. 4.7 Nuerical Methods A sequece of sies ad cosies ca be coputed very rapidly two ultiplicatios, oe additio each ad accurately with the followig recursio relatios: Let s = si ad c = cos. Defie s = siθ ad c = cosθ, the s = siθ + ca be coputed quickly, alog with c = cosθ +, by usig s = cs + sc ad c = cc ss. This ethod looses less tha 3 decials of floatig poit precisio after thousad iteratios. 5. Expoetial Itegral 5..5-a E x = x! [ = x E x e ]! x + This is related to 5.. ad but is easier to work with tha either; ad is uerically ore stable a Special Values E = A hady-dady value to have aroud

3 5..5-a Add Note: See also a Asyptotic Expasio The hypercoverget ca be obtaied fro the foral Euler Su =!w + = e x/w x dx 5..5-b For values of egative, see Gaa Fuctio 6..-a The followig itegrals look siilar but are i fact very differet: t z e t dt See Riea Zeta, sectio 3. + t z e t dt See Debye Fuctio, sectio 7. For iteger z, see b Γ + ε = e x x ε dx = + ε e x lxdx = εγ For sall, real ε >. Matheatically trivial, but hady if you just wated to look up this itegral. 6..-a k k = { f or x < where Θx = f or x = k + k + Θ k } is the Heaviside step fuctio. 3

4 6.3.-a x lxdx = / + for Aother hady itegral deservig etio 6.3.-b t z lt dt = ψ ψ z + z + ad for iteger z = we have [ t lt dt = ] + + Aother hady dady itegral to have aroud a x Γ,x = Γe = x for iteger. See also 5..8! This is a special case that should be etioed explicitly a S u e x + xu u k! dx = k! = u e/u Γ +, u Occurs i certai types of stochastic equatios; uerically upleasat to evaluate See also Suatio of Ratioal Series Sectio 6.8 should really be broke out ito its ow, ad fortified with various utilitaria sus, e.g. the below. Sus occur i ay probles, ad should get a hady referece chapter, aalogous to chapter 3, o their ow.. 4

5 6.8. = + z = π si πz See = + z =! ψ z See = + z + z + = + z 6.9 Foral Sus, Spectral Asyetries Soe forally diverget sus ca be give eaigful values through regularizatio. For exaple, li t k k + e tk = 4 ad thus we write, forally, k k + = 4 with the uderstadig that regulatio took place. This is because other regulators, besides e tk ca be used: for exaple, e t k provides excellet uerical stability, while i the liit s is better suited to aalytical treatets. Geeral theories of series ks acceleratio ca be applied o forally diverget sus to get eaigful results s = = 6.9. = + s = s = + + s = s + 5

6 6.9.4 = s = s = p + s = p!s + p s... p = = = + s Follows fro above, & etc. = ss + 6. Fiite Sus I copied these sus fro soe other book; they belog here. 6.. k= k 4 = [ ]/3 6.. k= k 5 = [ + + ]/ 6..3 k= k 6 = [ ]/4 6

7 6..4 k= k 7 = [ ]/ k= k = 6..6 k= k = 4 / k= k 3 = 6..8 k= kk + = [ ]/ 6. Diverget Sus Forally diverget sus that ca be writte as liits of coverget sus. 6.. li t k e tk = 6.. li t k k + e tk = 3 4 7

8 6..3 li t k k + k + 3e tk = li t k k + k + 3k + 4e tk = li t k k + k + 3k + 4k + 5e tk = These are readily obtaied[] by cosiderig the bioial geeratig fuctio. That is, defie A x = Γk + + x k Γk + Γ + = x Γ + = x k= k x k + x + ad so the above sus are give by + A li A x = Γ + x + 7. Error Fuctio 7..4-a Itegral Represetatios er f z = z e t π e z t dt See also z 8

9 7. Repeated Itegrals i, rather, i stads for itegral. Usig i to stad for itegral was a poor choice of otatio for this etire sectio Repeated Itegrals, Recurrece Relatios Let I z = I tdt be the idefiite itegral of erf, that is, I z = e t dt π the I z = z I z + I z z! This looks like 7..5 but is the erf=-erfc versio of that relatio. The etire sectio 7. should be redoe with erf ad erfc versios of the repeated itegral. 7.4-a Defiite ad Idefiite Itegrals z [er f ct] e t dt = π [er f cz]+ + Just aother hady itegral 7.4-b a e a z π er f cbzdz = x er f caxer f cbx b er f caze b z dz x Sadly, there s o closed for for this beastie. 7.4-c z a t e a t er f cbtdt = b π a + b e a +b z ze a z er f cbz 7.4-d May of the itegrals i sectio 7.4 ca be obtaied by writig ad the doig the x itegral first. f xer f zxdx = dx f x x dze z x 9

10 . Bessel Fuctios of Fractioal Order..4-a Asyptotic Expasios For x real, x, j x = x six π/ + O x y x = cosx π/ + O x x..4-b For fixed, real x ad ex j x + By use of Sterlig s forula...-a Asyptotic Expasios f z = / z f z = + O z 3 for eve, positive or egative, ad + / z + O z 4 for odd, positive or egative...-b Thus, for k eve, k we have j k z = k/ z siz+ k/ ad, for k odd, k we have j k z = k+/ z k k + z cosz- k+/ k k + z cosz + O z 3 siz + O z 3 Although, see..4-a above for the correct treatet of the asyptotic phase agle. The phase agle is eeded for quatu scatterig probles.

11 . Itegrals of Bessel Fuctios. Siple Itegrals of Bessel Fuctios The z liit of these itegrals is o-trivial. See.4.6, a tjν tdt = z [ J ν z J ν zj ν z ] for Rν > This closed for is easier to work with tha the ifiite su give, ad also reduces the order o the RHS..3.3-b ν Jν tj ν t dt = z [ Jν z J ν+ zj ν z ] + J ν zj ν z t.3.34-a Special case of a See b J ν tj ν+ tdt = = J ν++ z Ulike.3.35, ν eed ot be iteger i this forula.3.36-a Cojecture: Itegrals of the type t J ξ tj ξ + tdt are solvable i closed for oly for + odd. Disproof of this cojecture would brig a iportat additio to this subsectio. Itegrals of the above for ca be attacked usig the recursio relatios J ν z = ν z J ν z + J ν z ad J ν+ z = ν z J ν z J ν z. A useful set of itegral recursio relatios, suitable for ueric evaluatio, are preseted below.

12 .3.36-b tjν tj ν+ tdt = ν ν zj ν z + ν + tjν tj ν tdt ν.3.36-c tjν tj ν+ tdt = z J ν z + ν + J ν tdt.3.36-d tjν tj ν+ tdt = zjν z + J ν tdt + tjν tj ν tdt.3.36-e J ν tdt = J ν zj ν z + J ν tdt Jν tj ν t dt t.3.36-f tj ν tj ν+ tdt = z J ν zj ν+ z + t Jν+ tdt J ν tdt.3.4-a Itegrals of Spherical Bessel Fuctios These occur i calculatios of wave fuctios ad are useful eough to deserve their ow sectio. The j zare spherical Bessel fuctios, of chapter..3.4-b µ + ν + t µ j ν tdt = Γ π µ for Rµ + ν > ad Rµ < ν µ + Γ.3.4-c t j t j t j + tdt = z j z follows fro.3.33

13 .3.4-d t [ j t j +t ] dt = z j z j + z.3.4-e t 3 j t j tdt = z3 4 [ + j z j + z j z ] 3. Beroulli ad Euler Polyoials - Riea Zeta Fuctio 3..3-a B = + + Σ k B k Hady for geeratig large B uerically a γ is Euler s costat, see γ i are called the Stieltjes costats. The first few are γ = ad γ =.9693 ad γ 3 =.5383 ad γ 4 = Sus of Riea Zeta Fuctios This is a ew sectio, ot i the curret A&S. Turs out these are a special case of I the below, ν ca be ay coplex value, ot ecessarily iteger ζ ν + = k + ν + k [ζ k + ν + ] 3.3. k + ν + k + [ζ k + ν + ] = See also for iteger ν. 3

14 k k + ν + k + k k + ν + k + k k + ν + k [ζ k + ν + ] = ν+ [ζ k + ν + ] = ν[ζ ν + ] ν [ζ k + ν + ] = ζ ν + ν S. p + p p= [ζ p + + ] = [ + k= k ζ k + ] For iteger Note that S = ad S = ζ ad S = ζ +ζ 3 ad i geeral S +S + = ζ +, which is to be used i Note li S = which is uerically satisfied for > T p + p p= [ζ p + + ] = + [ + ζ + k= k kζ k + For iteger. This follows fro the observatio T + T + = S whe used i ] The above trick ca be repeated to express fiite su, for ay iteger k. p= p + k p [ζ p + + ] as a For iteger >, k k + ν + k ζ k + ν + = j= j j ζ ν + j 4

15 3.3. k k + ζ k + = = k= k + = = ψ k + ζ k + = = k= k = = ψ Note this is a foral diverget su that ca be ade eaigful through regularizatio. XXX Need to do this. Refereces [] Stephe Crowley. Notes about the gkw operator. persoal couicatio,. 5