The integrals in Gradshteyn and Ryzhik. Part 13: Trigonometric forms of the beta function
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1 SCIENTIA Series A: Mathematical Scieces, Vol 19 (1, Uiversidad Técica Federico Sata María Valparaíso, Chile ISSN c Uiversidad Técica Federico Sata María 1 The itegrals i Gradshtey ad Ryzhik Part 13: Trigoometric forms of the beta fuctio Victor H Moll Abstract The table of Gradshtey ad Ryzhik cotais some trigoometric itegrals that ca be expressed i terms of the beta fuctio We describe the evaluatio of some of them 1 Itroductio The table of itegrals [] cotais a large variety of defiite itegrals i trigoometric form that ca be evaluated i terms of the beta fuctio defied by (11 B(a, b = x a 1 (1 x b 1 dx The covergece of the itegral requires a, b > The chage of variables x = si t yields the basic represetatio (1 B(a, b = that, after replacig (a, b by (a, b, is writte as (13 This appears as 3615 i [] (1 si a 1 t cos b 1 t dt, si a 1 t cos b 1 t dt = 1 B ( a, b Special cases I this sectio we preset several special cases of formula (13 that appear i [] Example 1 The choice b = 1 i (13 gives si a 1 t dt = 1 B ( a, 1 Mathematics Subject Classificatio Primary 33 Key words ad phrases Itegrals, Beta fuctio The author wishes to ackowledge the partial support of NSF-DMS
2 9 V MOLL Legedre s duplicatio formula ( Γ(a = a 1 Γ(aΓ(a + 1 ca be used to write (1 as ( a (3 si a 1 t dt = a B, a This is 3611 i [] The dual evaluatio ( a (4 cos a 1 t dt = a B, a = a Γ (a/ Γ(a = a Γ (a/, Γ(a comes from the chage of variables t t The reader will fid a proof of ( i [1] Example The special case a = 1 i (3 gives 3617: dx ( (5 = Γ 1 4 si x Example 3 The special case a = 3 i (3 gives 3616: (6 ( si xdx = Γ 1 4 (7 (8 Example 4 The special case a = 5 i (3 gives 361: si 3/ xdx = 1 6 Γ ( 1 4 Example 5 The special case a = m + 1 i (3 gives ad usig the idetity (9 Γ ( m + 1 it yields (1 si m xdx = m 1 B ( m + 1, m + 1, (m! = m m! si m xdx = ( m m m+1 This appears as 3613 Similarly, a = m + i (3 gives (11 that ca be writte as (1 This is 3614 si m+1 xdx = m B(m + 1, m + 1, si m+1 xdx = ( 1 m m (m + 1 m
3 TRIGONOMETRIC BETA INTEGRALS 93 Example 6 The itegral 361 is ta ±a xdx = si ±a x cos a xdx = 1 B ( 1±a, 1 1±a = 1 Γ ( ( 1±a Γ 1 1±a ad this reduces to ta ±a xdx = as it appears i the table Example 7 The idetity cos(a/, (13 ta a 1 x cos b x = si a 1 x cos b a 1 x shows that (14 ta a 1 x cos b xdx = This appears as 3631 si a 1 x cos b a 1 xdx = 1 B ( a, b a Example 8 The formula 364 states that si a 1/ x (15 cos a 1 x dx = Γ ( a Γ(1 a Γ ( 5 4 a This comes directly from (13 (16 Example 9 The idetity 367: ta a x cos a x dx = cot a x si a x dx = Γ(aΓ(1 a ( a a si, ca be verified by writig the first itegral as (17 I = si a x cos 1 a xdx = 1 ( a + 1 B, 1 a The beta fuctio is ( 1 a + 1 (18 B, 1 a = Γ ( a + ( 1 Γ 1 a Γ ( 1 a Usig Γ(tΓ(1 t = we ca reduce (18 to the expressio i (16 (19 si t Example 1 The evaluatio of 368 is direct, oce we write the itegral as ( sec p x si p 1 xdx = Γ(pΓ(1 p, cos p xsi p 1 xdx = 1 B ( 1 p, p
4 94 V MOLL 3 A family of trigoometric itegrals I this sectio we preset the evaluatio of a family of trigoometrical itegrals i [] May special cases appear i the table Propositio 31 Let a, b, c R with the coditio (31 a + b + c + = The (3 (33 Proof Let t = ta x to obtai ad (31 yields (34 si a x cos b x cos c (xdx = 1 ( a + 1 B, c + 1 si a x cos b x cos c (xdx = si a x cos b x cos c (xdx = The chage of variables s = t produces (35 si a x cos b x cos c (xdx = 1 ad this last itegral has the give beta value t a (1 t c (1 + t (a+b+c+/ dt t a (1 t c dt s (a 1/ (1 s c ds, Example 3 The formula (3, with a =, b = p ad c = p appears as 365 i []: (36 si x cos p (x cos p++ x dx = 1 B ( + 1, p + 1 Example 33 The formula 3643 cos 1/ (x (37 cos +1 x dx = ( +1 correspods to the case a =, b = 1 ad c = 1 (38 Example 34 Formula 3644 i [] cos µ (x cos (µ+1 x dx = µ B(µ + 1, µ + 1 correspods to a =, b = µ ad c = µ The (3 gives cos µ (x (39 cos (µ+1 x dx = 1 ( 1 B, µ + 1 The duplicatio formula (31 Γ(x = x 1 Γ(xΓ(x + 1,
5 TRIGONOMETRIC BETA INTEGRALS 95 trasforms (39 ito (38 (311 Example 35 The values a = µ, b = ad c = µ produce 3645: si µ x cos µ (x dx = Γ(µ 1 Γ(1 µ directly Ideed, the aswer from (3 is B(µ 1/, 1 µ/ The table also has the alterative aswer 1 µ B(µ 1, 1 µ that ca be obtaied usig (31 (31 Example 36 Formula 3651: si 1 x cos p (x cos p++ x dx = 1 B(, p + 1 correspods to a = 1, b = p 1 ad c = p (313 Example 37 The choice a = 1, b = m ad c = m 1 gives 3653: For, m N we ca also write (314 (315 si 1 x cos m 1/ (x cos +m x si 1 x cos m 1/ (x cos +m x dx = 1 dx = 1 B(, m + 1 ( m m ( + m + m 1 ( 1 + m Example 38 The values a =, b = m 1 ad c = m 1 give 3654: For, m N we ca also write (316 (317 si x cos m 1/ (x cos +m+1 x si x cos m 1/ (x cos +m+1 x Example 39 Formula 3661: dx = dx = 1 B ( + 1, m + 1 +m+1 ( ( m m si 1 x 1 cos(x dx = cos + B(, 3/, x ( 1 + m comes from (3 with a = 1, b = ad c = 1/ For N we have (318 (319 si 1 x ( 1!! cos(x dx = cos + x ( + 1! Example 31 The last example i this sectio is formula 366: si x 1 cos(xdx = cos +3 x B( + 1, 3, comes from (3 with a =, b = 3 ad c = 1/ For N we have (3 si x cos(x dx = cos +3 x + (!! ( + 1!
6 96 V MOLL Refereces [1] G Boros ad V Moll Irresistible Itegrals Cambridge Uiversity Press, New York, 1st editio, 4 [] I S Gradshtey ad I M Ryzhik Table of Itegrals, Series, ad Products Edited by A Jeffrey ad D Zwilliger Academic Press, New York, 7th editio, 7 Departmet of Mathematics, Tulae Uiversity, New Orleas, LA 7118, USA address: vhm@mathtulaeedu Received 7 7 9, revised 1 1 1
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