HermiteHGeeral Notatios Traditioal ame Hermite fuctio Traditioal otatio H Mathematica StadardForm otatio HermiteH, Primary defiitio 07.0.0.000.0 H F ; ; F ; 3 ; Specific values Specialied values For fixed H 0 For fixed 07.0.03.000.0 Explicit iteger 07.0.03.000.0 H 0 07.0.03.0003.0 H 07.0.03.0004.0 H 4
http://fuctios.wolfram.com 07.0.03.0005.0 H 3 8 3 07.0.03.0006.0 H 4 48 6 4 07.0.03.0007.0 H 5 0 60 3 3 5 07.0.03.0008.0 H 6 0 70 480 4 64 6 07.0.03.0009.0 H 7 680 3360 3 344 5 8 7 07.0.03.000.0 H 8 680 3 440 3 440 4 3584 6 56 8 07.0.03.00.0 H 9 30 40 80 640 3 48 384 5 96 7 5 9 07.0.03.00.0 H 0 30 40 30 400 403 00 4 6 80 6 3 040 8 04 0 Symbolic iteger 07.0.03.003.0 H k k k k ; 07.0.03.004.0 H H k k k L k k k j j erf k j 0 k 3 j L j j j 0 3 4 j 4 ; Geeral characteristics Domai ad aalyticity H is a aalytical fuctio of ad which is defied over. For fixed, it is a etire fuctio of. For fixed, it is a etire fuctio of. For iteger, H degeerates to a polyomial i. 07.0.04.000.0 H
http://fuctios.wolfram.com 3 Symmetries ad periodicities Parity 07.0.04.000.0 H H ; Mirror symmetry H H Periodicity No periodicity 07.0.04.0003.0 Poles ad essetial sigularities With respect to For fixed ;, the fuctio H has oly oe sigular poit at. It is a essetial sigular poit. 07.0.04.0004.0 ig H, ; For positive iteger, the fuctio H is polyomial ad has pole of order at. 07.0.04.0005.0 ig H, ; With respect to For fixed, the fuctio H has oly oe sigular poit at. It is a essetial sigular poit. 07.0.04.0006.0 ig H, Brach poits With respect to For fixed, the fuctio H does ot have brach poits. 07.0.04.0007.0 H With respect to For fixed, the fuctio H does ot have brach poits. 07.0.04.0008.0 H Brach cuts With respect to
http://fuctios.wolfram.com 4 For fixed iteger, the fuctio H does ot have brach cuts. 07.0.04.0009.0 H With respect to For fixed, the fuctio H does ot have brach cuts. 07.0.04.000.0 H Series represetatios Geeralied power series Expasios at geeric poit 0 For the fuctio itself 07.0.06.000.0 H H 0 H 0 0 H 0 0 ; 0 07.0.06.00.0 H H 0 H 0 0 H 0 0 O 0 3 H k 07.0.06.00.0 k k 07.0.06.003.0 H k 0 0 k k k H k 0 F, ; k, k k k ; 0 0 F, ; k, 3 k ; 0 0 k 07.0.06.004.0 H H 0 O 0 Expasios at 0 For the fuctio itself Geeral case H 07.0.06.000.0 4 3 6 3 4 ; 0 30
http://fuctios.wolfram.com 5 H H 07.0.06.005.0 07.0.06.000.0 07.0.06.0003.0 4 O 6 3 6 3 4 O 6 30 k k k k k k 3 k k H F ; ; F ; 3 ; 07.0.06.0004.0 H O O 07.0.06.006.0 H F, ; F m, m k k k k m k k 3 k k m H m m m F, m ; m 3, m ; m 3 m F, m 3 ; m, m 5 ; m Summed form of the trucated series expasio. Special cases 07.0.06.0005.0 H k k k k ; 07.0.06.007.0 H O ; Expasios at For the fuctio itself Special cases
http://fuctios.wolfram.com 6 07.0.06.008.0 3 H ; 4 3 4 07.0.06.009.0 3 H O ; 4 3 4 6 07.0.06.000.0 H k k 07.0.06.00.0 k k k ; H k k k k ; 07.0.06.00.0 H F 0, 07.0.06.003.0 H O ; ; ; ; Asymptotic series expasios 07.0.06.004.0 si H 4 csc sec 3 4 3 4 3 4 ; 4 3 4 H 07.0.06.005.0 3 O arg 4 3 4 6 3 O 4 3 4 6 3 O 4 3 4 6 ; 3 4 O arg 4 3 4 6 3 4 O True 4 3 4 6
http://fuctios.wolfram.com 7 07.0.06.006.0 si H 4 csc sec k k k k k O k k k k O ; 07.0.06.007.0 k k k k k O arg H k k k k k O k k k k O arg ; k k k k k O k k k k O True 07.0.06.0006.0 si H F 0, ; ; csc sec 4 F 0, ; ; ; H 07.0.06.008.0 F 0, ; ; arg F 0, ; ; F 0, ; ; arg F 0, ; ; F 0, ; ; True ; 07.0.06.0007.0 H O 4 cos si O ; H 07.0.06.009.0 O arg O O arg O O True ; 07.0.06.0008.0 H F 0, ; ; ; arg
http://fuctios.wolfram.com 8 07.0.06.0009.0 H O ; arg Itegral represetatios O the real axis H 07.0.07.000.0 0 t t cos t 07.0.07.000.0 H 3 t t t ; t ; Re Itegral represetatios of egative iteger order Rodrigues-type formula. 07.0.07.0003.0 H ; Limit represetatios 07.0.09.000.0 H lim Λ L Λ Λ Λ Λ 07.0.09.000.0 H lim Λ Λ C Λ Λ ; 07.0.09.0003.0 H lim a a,a P a a Geeratig fuctios 07.0..000.0 H t tt ; Differetial equatios Ordiary liear differetial equatios ad wroskias 07.0.3.0005.0 w w w 0 ; w c H c H
http://fuctios.wolfram.com 9 07.0.3.0006.0 W H, H 07.0.3.0007.0 w w w 0 ; w c H c F ; 3 ; F ; ; 07.0.3.0008.0 W H, F ; 3 ; F ; ; 07.0.3.000.0 w w w 0 ; w c H c F ; ; 07.0.3.000.0 W H, F ; ; 07.0.3.0003.0 w w w 0 ; w c H c F ; 3 ; W H, F 07.0.3.0004.0 ; 3 ; 07.0.3.0009.0 w g g g g w g w 0 ; w c H g c g H g 07.0.3.000.0 W H g, g H g g g 07.0.3.00.0 w g g h h g g w w c h H g c h g H g 07.0.3.00.0 g g h g h W h H g, h g H g g h g h g h g h h h h w 0 ; 07.0.3.003.0 w a r r r s w a r s r r s r s w 0 ; w c s H a r c s a r H a r 07.0.3.004.0 W s H a r, s a r H a r a a r r s r
http://fuctios.wolfram.com 0 07.0.3.005.0 w a r logr logs w a logr logr logs r logs logr logs w 0 ; w c s H a r c s a r H a r 07.0.3.006.0 W s H a r, s a r H a r a a r r s logr Trasformatios Trasformatios ad argumet simplificatios Argumet ivolvig basic arithmetic operatios 07.0.6.0007.0 H H ; arg 07.0.6.0008.0 H H si L 07.0.6.0009.0 H H F ; 3 ; 07.0.6.000.0 H H ; 07.0.6.00.0 H H si L 07.0.6.00.0 H H F 07.0.6.000.0 H H ; ; 3 ; Additio formulas 07.0.6.003.0 k H k H k 07.0.6.004.0 k k k H H k k
http://fuctios.wolfram.com 07.0.6.000.0 H k H k H k ; 07.0.6.005.0 H cosα siα cos k taα k Α H k H k k 07.0.6.0003.0 cos k Α si k Α H k H k H cosα siα ; k k Multiple argumets 07.0.6.0004.0 H k k k H k ; Products, sums, ad powers of the direct fuctio Products of the direct fuctio 07.0.6.0005.0 mi,m k H km H H m m ; m k k m k 07.0.6.0006.0 mi,m H H m m k k H km k k ; m Idetities Recurrece idetities Cosecutive eighbors 07.0.7.000.0 H H H 07.0.7.000.0 H H H Distat eighbors 07.0.7.0006.0 H, H, H ; 0,,,,,
http://fuctios.wolfram.com 07.0.7.0007.0 H, H, H ; 0,,,,, Fuctioal idetities Relatios betwee cotiguous fuctios Recurrece relatios 07.0.7.0003.0 H H H 07.0.7.0004.0 H H H Normalied recurrece relatio 07.0.7.0005.0 p, p, p, ; p, H Complex characteristics Real part 07.0.9.000.0 j j j y j ReH x y j 0 H j x ; x y j Imagiary part 07.0.9.000.0 j j j y j ImH x y H j x ; x y j j 0 Differetiatio Low-order differetiatio With respect to
http://fuctios.wolfram.com 3 07.0.0.000.0 H log H 4 Ψ 6 3 F 0 F ; ; ; ;, ;, 3 ;; ;, F 0 Ψ F 3 ; ;, ;,, 5 3 ;; ; ; 3 ; 07.0.0.000.0 H log H 4 Ψ F ; ; Ψ F ; 3 ; Ψ F ; Ψk ; k k k k Ψ F ; 3 ; Ψk k k k 3 k 07.0.0.0003.0 H log H Ψ log6 Ψ Ψ F ; ; Ψ log6 Ψ Ψ F ; 3 ; log4 Ψ Ψ F ; ; k Ψk k k k Ψ Ψ F ; ; k k k k Ψ k Ψ Ψ k Ψ k 4 log4 Ψ Ψ F ; 3 ; Ψk k k k 3 k Ψ Ψ F ; 3 ; k k k 3 k Ψ k Ψ Ψ k Ψ k
http://fuctios.wolfram.com 4 07.0.0.00.0 H,0 F, ; 3, ; erfi Ψ H Ψ k 4k k k k k L k k k k k L k L k ; Brychkov Yu.A. (006) 07.0.0.00.0 H,0 0 F, ; 3, ; erfi Brychkov Yu.A. (006) 07.0.0.003.0 H,0 F, ; 3, ; erfc erfi Brychkov Yu.A. (006) H 0 07.0.0.004.0 With respect to log4ψ 07.0.0.0004.0 H H 07.0.0.0005.0 H 4 H Backward shift operator: 07.0.0.0006.0 H H H 07.0.0.0007.0 0 True H H Symbolic differetiatio With respect to
http://fuctios.wolfram.com 5 m H 07.0.0.005.0 m m m k log mk j 0 k j k j k j j 0 j k j j ; m With respect to 07.0.0.0008.0 m H m m m H m ; m 07.0.0.0009.0 m H m m m F, ; m, m ; m m F, ; m, 3 m ; ; m Fractioal itegro-differetiatio With respect to 07.0.0.000.0 Α H Α Α Α F, ; Α, Α ; Α Α F, ; Α, 3 Α ; Itegratio Idefiite itegratio Ivolvig oly oe direct fuctio 07.0..000.0 H H Ivolvig oe direct fuctio ad elemetary fuctios Ivolvig power fuctio 07.0..000.0 Α H Α Α F, Α ;, Α ; Α F, Α ; 3, Α 3 ; 07.0..0003.0 H 3 3 F ; ; 4 3 F ; 5 ;
http://fuctios.wolfram.com 6 07.0..0004.0 m H a m m F, m a a m F, m ; 3, m ; a 07.0..0005.0 3 H F ; ; F ; 3 ; Ivolvig expoetial fuctio 07.0..0006.0 H F ; 3 ; F ; ; Ivolvig expoetial fuctio ad a power fuctio 07.0..0007.0 Α p H 07.0..0008.0 Α p H 07.0..0009.0 Α a H a Α p Α Α Α p k Α, p k 3 k k p k Α k, p k 3 k pk k Α p Α Α p Α k Α, p k k k p k k Α k, p k pk k Α Α F 07.0..000.0, Α ;, Α a Α ; a F H si F Ivolvig fuctios of the direct fuctio ad elemetary fuctios, Α ; 3, Α 3 ; a ; 3 ; F ; ; Ivolvig elemetary fuctios of the direct fuctio ad elemetary fuctios Ivolvig powers of the direct fuctio ad a power fuctio 07.0..00.0 H 8 H H
http://fuctios.wolfram.com 7 Ivolvig powers of the direct fuctio, power ad expoetial fuctios 07.0..00.0 H 4 H H Ivolvig direct fuctio ad Gamma-, Beta-, Erf-type fuctios Ivolvig probability itegral-type fuctios Ivolvig erf 07.0..003.0 erfa H a a a H a erfa H a Ivolvig erfi 07.0..004.0 erfi H H erfi H Defiite itegratio Ivolvig the direct fuctio 07.0..005.0 t Α a t H tt a Α 0 Α F Α, ; ; a Α a Α a F Α, ; ; a a Α Α F, ; ; a ; Rea 0 ReΑ 0 07.0..006.0 t H tt ; Orthogoality: 07.0..007.0 t H m t H tt,m ; m 07.0..008.0 t H l t H m t H tt lm l m l m l m l m ; l m l m l m m l l m
http://fuctios.wolfram.com 8 07.0..009.0 t H l t H m t H tt 0 ; l m l m l m m l l m Summatio Fiite summatio 07.0.3.000.0 H k x H k y H x H y H x H y ; k k x y k k k 07.0.3.000.0 k H k x H k y x y ; 07.0.3.0003.0 cos k H k x H k y Rex y ; 07.0.3.0004.0 si k H k x H k y Imx y ; Ifiite summatio 07.0.3.0005.0 H w ww 0 07.0.3.0006.0 H w w cos 0 w 07.0.3.0007.0 H w w si 0 w w 07.0.3.0008.0 c H w w c F c; ; w w 0 07.0.3.0009.0 c H w 0 w c F c ; 3 ; w w H w 0 07.0.3.000.0 4 w w 4 w 3 exp 4 w 4 w
http://fuctios.wolfram.com 9 07.0.3.00.0 H x y t x ΑΑ W t y ; Α t y y Α y 0 07.0.3.00.0 H kj w kj j 0 k j j l l 4 l exp j w j w k l j ; k j 07.0.3.003.0 H H w 0 4 w exp w w 4 w ; w 07.0.3.004.0 H x H y 0 x y x y ; x y 0 07.0.3.005.0 m H H m m H m ; m Operatios Limit operatio 07.0.5.000.0 lim 4 H si ; lim 4 07.0.5.000.0 H cos ; Orthogoality, completeess, ad Fourier expasios The set of fuctios H x, 0,,, forms a complete, orthogoal (with weight iterval,. 07.0.5.0003.0 x ) system o the 0 x H x y H y x y 07.0.5.0004.0 m m t H m t t H t t,m
http://fuctios.wolfram.com 0 Ay sufficietly smooth fuctio f x ca be expaded i the system H x 0,, as a geeralied Fourier series, with its sum covergig to f x almost everywhere. 07.0.5.0005.0 f x c Ψ x ; c Ψ t f tt Ψ x 0 x H x ; x Represetatios through more geeral fuctios Through hypergeometric fuctios Ivolvig F 07.0.6.000.0 H F ; ; F ; 3 ; Ivolvig F 07.0.6.000.0 H F ; ; F ; 3 ; Ivolvig p F q 07.0.6.0003.0 H F 0, Ivolvig hypergeometric U 07.0.6.0004.0 ; ; ; H U,, ; Re 0 07.0.6.0005.0 H U, 3, ; Re 0 07.0.6.0006.0 H F ; ; U,, 07.0.6.0007.0 H U, 3, F ; ; H 07.0.6.0008.0 F ; 3 ; U,,
http://fuctios.wolfram.com H 07.0.6.0009.0 F ; 3 ; U, 3, Through Meijer G Classical cases for the direct fuctio itself H 07.0.6.009.0 G,, 0, ; 07.0.6.00.0 H 3 G, 3,4, 8, 5 8 0,, 8, 5 8 07.0.6.00.0 H G,, 0, ; arg 07.0.6.003.0 H H 3 G,,3, 4 0,, 4 07.0.6.004.0 H H 3 G,,3, 4, 0, 4 Classical cases ivolvig exp 07.0.6.000.0 H G,0, 0, ; arg 07.0.6.00.0 H G,0, 0, 07.0.6.005.0 H G,,3, 0,, 07.0.6.006.0 H H cos G,, 0,
http://fuctios.wolfram.com 07.0.6.007.0 H H si G,,, 0 Classical cases ivolvig exp ad cosh 07.0.6.008.0 cosh H G,0, 0, G,, 4 0, ; arg 07.0.6.009.0 cosh, H G,,3 0,, 3,, 5 G, 3,4 8 8 0,,, 5 8 8 ; arg cosh 07.0.6.0030.0 H H cos G,, 0, 3, G,,3 4 0,, 4 ; arg 07.0.6.003.0 cosh H H 3, G,,3 4, 0, 4 G,, si, 0 ; arg 07.0.6.003.0 cosh H G,0, 0, G,, 4 0, 07.0.6.0033.0 cosh H G,,3, 0,, 3 G, 3,4, 8, 5 8 0,, 8, 5 8 07.0.6.0034.0 cosh H H cos G,, 0, 3 G,,3, 4 0,, 4 07.0.6.0035.0 cosh H H 3 G,,3, 4, 0, 4 G,, si, 0 Classical cases ivolvig exp ad sih 07.0.6.0036.0 sih H 4 G,, 0, G,0, 0, ; arg
http://fuctios.wolfram.com 3 07.0.6.0037.0 sih H 3,, 5 G, 3,4 8 8 0,,, 5 8 8, G,,3 0,, ; arg 07.0.6.0038.0 sih H H 3, G,,3 4 0,, 4 cos G,, 0, ; arg 07.0.6.0039.0 sih H H 3, G,,3 4, 0, 4 si G,, ;, 0 arg sih H 07.0.6.0040.0 G,, 4 0, G,0, 0, 07.0.6.004.0 sih H 3 G, 3,4, 8, 5 8 0,, 8, 5 8 G,,3, 0,, 07.0.6.004.0 sih H H 3 G,,3, 4 0,, 4 cos G,, 0, 07.0.6.0043.0 sih H H 3 G,,3, 4, 0, 4 si G,,, 0 Classical cases for products of H 07.0.6.0044.0 H 4 4 H 4 4 3 G 4,,4 4, 0, 4,, 3 4 Classical cases ivolvig Exp ad products of H 4 H 07.0.6.0045.0 H 4 3 4,0 G,4 4, 0, 4,, 3 4 Classical cases ivolvig Exp ad parabolic cylider D 07.0.6.0046.0 D H 3 G 4,,4 4 4, 0, 4,, 3 4 ; arg 0
http://fuctios.wolfram.com 4 07.0.6.0047.0 D H 4,0 G 4,4 4, 0, 4,, 3 4 ; 4 arg 4 Geeralied cases for the direct fuctio itself 07.0.6.000.0 H G,,, 0, ; 07.0.6.00.0 H G,,, 3, ; H 07.0.6.0048.0 3 G, 3,4,, 8, 5 8 0,, 8, 5 8 07.0.6.003.0 H G,0,,, ; 07.0.6.004.0 H lim m m G,,, m3, ; 07.0.6.0049.0 H H 3 G,,3,, 4 0,, 4 07.0.6.0050.0 H H 3 G,,3,, 4, 0, 4 Geeralied cases ivolvig exp 07.0.6.005.0 H G,0,, 0, 07.0.6.005.0 H G,,3,, 0,, 07.0.6.005.0 H H cos G,,, 0,
http://fuctios.wolfram.com 5 07.0.6.0053.0 H H si G,,,, 0 Geeralied cases ivolvig exp ad cosh 07.0.6.0054.0 cosh H G,0,, 0, G,,, 4 0, 07.0.6.0055.0 cosh H G,,3,, 0,, 3 G, 3,4,, 8, 5 8 0,, 8, 5 8 07.0.6.0056.0 cosh H H cos G,,, 0, 3 G,,3,, 4 0,, 4 07.0.6.0057.0 cosh H H 3 G,,3,, 4, 0, 4 si G,,,, 0 Geeralied cases ivolvig exp ad sih 07.0.6.0058.0 sih H 4 G,,, 0, G,0,, 0, 07.0.6.0059.0 sih H 3 G, 3,4,, 8, 5 8 0,, 8, 5 8 G,,3,, 0,, 07.0.6.0060.0 sih H H 3 G,,3,, 4 0,, 4 cos G,,, 0, 07.0.6.006.0 sih H H 3 G,,3,, 4, 0, 4 si G,,,, 0 Geeralied cases for products of H 07.0.6.006.0 3 H 4 H 4 4, G,4, 4, 0, 4,, 3 4 Geeralied cases ivolvig Exp ad products of H
http://fuctios.wolfram.com 6 07.0.6.0063.0 H H 4,0 G,4,, 0, 4,, 3 4 07.0.6.0064.0 H H 4,0 G,4, 4, 0, 4,, 3 4 Geeralied cases ivolvig Exp ad parabolic cylider D 07.0.6.0065.0 D H 3 G 4,,4 4, 4, 0, 4,, 3 4 07.0.6.0066.0 D H 4,0 G,4, 4, 0, 4,, 3 4 Through other fuctios Ivolvig some hypergeometric-type fuctios 07.0.6.0067.0 H cos L si L 07.0.6.0068.0 H L ; 07.0.6.0069.0 H L ; 07.0.6.006.0 H lim Λ L Λ Λ Λ Λ 07.0.6.007.0 H lim Λ Λ C Λ Λ ; 07.0.6.008.0 H lim a a,a P a a Represetatios through equivalet fuctios With related fuctios
http://fuctios.wolfram.com 7 07.0.7.000.0 H cos L si 07.0.7.000.0 H L ; L 07.0.7.0003.0 H L ; 07.0.7.0004.0 H D Zeros 07.0.30.000.0 x j ; H x k 0 x k j x k k j Theorems Expasios i geeralied Fourier series f x c k Ψ k x ; c k f tψ k tt, Ψ k x k 4 k x H k x, k. Fourier trasform eigefuctios Hermite polyomials together with their weightig fuctio are eigefuctios of the Fourier ad iverse Fourier trasforms: t x x H x x t H t ; Zeros of Hermite polyomials For ay give iterval a, b, there exists some such that H x has a ero i this iterval. The umber of simple graphs The umber of simple graphs with o cycles ad vertices is H H. History
http://fuctios.wolfram.com 8 P. S. Laplace (80) Ch. Hermite (864) N. J. Soie (880)
http://fuctios.wolfram.com 9 Copyright This documet was dowloaded from fuctios.wolfram.com, a comprehesive olie compedium of formulas ivolvig the special fuctios of mathematics. For a key to the otatios used here, see http://fuctios.wolfram.com/notatios/. Please cite this documet by referrig to the fuctios.wolfram.com page from which it was dowloaded, for example: http://fuctios.wolfram.com/costats/e/ To refer to a particular formula, cite fuctios.wolfram.com followed by the citatio umber. e.g.: http://fuctios.wolfram.com/0.03.03.000.0 This documet is curretly i a prelimiary form. If you have commets or suggestios, please email commets@fuctios.wolfram.com. 00-008, Wolfram Research, Ic.