Formulas and Identities of Inverse Hyperbolic Functions

Transcription

1 FORMALIZED MATHEMATICS Volume 13, Number 3, Pages Universit of Bia lstok, 005 Formulas and Identities of Inverse Hperbolic Functions Fuguo Ge Qingdao Universit of Science and Technolog Xiquan Liang Qingdao Universit of Science and Technolog Yuzhong Ding Qingdao Universit of Science and Technolog Summar. This article describes definitions of inverse hperbolic functions and their main properties, as well as some addition formulas with hperbolic functions. MML identifier: SIN COS7, version: The papers [1], [8], [4], [], [9], [3], [6], [5], and [7] provide the terminolog and notation for this paper. 1. Preliminaries In this paper,, t denote real numbers. Net we state a number of propositions: (1) If > 0, then 1 = 1. () If > 1, then ( 1 ) < 1. (3) ( +1 ) < 1. (4) + 1 > 0. (5) > c 005 Universit of Bia lstok ISSN

2 384 fuguo ge and iquan liang and uzhong ding (6) If 0 and 1, then +1 (7) If 0 and 1, then 1 +1 (8) If 1, then 1. (9) If 0 and 1, then (10) If 1, then + > 0. (11) If < 1, then + 1 > 0 and 1 > 0. (1) If 1, then (1 ) > 0. (13) If < 1, then +1 1 (14) If < 1, then ( 1+ ) < 1. (15) If 0 < and < 1, then 1+ 1 > 0. (16) If 0 < and < 1, then < 1. (17) If 0 < and < 1, then 1 1 > 1. (18) If 0 < and < 1, then 1 > 0. (19) If 0 < and < 1, then 0 < (1 ). (0) If 0 < and < 1, then 1+ 1 > 1. (1) If 1 <, then ( 1 ) < 1. () If 0 < and 1, then 1 (3) If 1, then 0 < + 1. (4) If 1 and 1, then (5) If 1 and 1 and, then 0 < 1. (6) If 1 and 1 and, then (7) If < 1 and < 1, then 1. (8) If < 1 and < 1, then 1. (9) If 0, then ep 1. (30) If 0, then (ep) 1 0. (31) If 0 < t, then t 1 t +1 < 1. (3) If 1 < t and t < 1, then 0 < t+1 1 t.. Formulas and Identities of Inverse Hperbolic Functions Let be a real number. The functor sinh ields a real number and is defined b: (Def. 1) sinh = log e ( Let be a real number. The functor cosh 1 ielding a real number is defined b:

3 formulas and identities of inverse (Def. ) cosh 1 = log e ( + 1 Let be a real number. The functor cosh ields a real number and is defined b: (Def. 3) cosh = log e ( + 1 Let be a real number. The functor tanh ields a real number and is defined b: (Def. 4) tanh = 1 log e( 1+ 1 Let be a real number. The functor coth ielding a real number is defined as follows: (Def. 5) coth = 1 log e( +1 1 Let be a real number. The functor sech 1 ields a real number and is defined b: (Def. 6) sech 1 = log e ( 1+ 1 Let be a real number. The functor sech ielding a real number is defined as follows: (Def. 7) sech = log e ( 1+ 1 Let be a real number. The functor csch ielding a real number is defined b: (Def. 8)(i) csch = log e ( ) if 0 <, (ii) csch = log e ( 1 1+ ) if < 0, (iii) < 0, otherwise. The following propositions are true: (33) If 0, then sinh = cosh (34) If < 0, then sinh = cosh + 1. (35) sinh = tanh ( +1 (36) If 1, then cosh 1 = sinh 1. (37) If > 1, then cosh 1 = tanh ( 1 (38) If 1, then cosh 1 = cosh 1 (39) If 1, then cosh = cosh (40) If 1, then cosh 1 = sinh. (41) If < 1, then tanh = sinh ( 1 (4) If 0 < and < 1, then tanh = cosh 1( 1 1 (43) If < 1, then tanh = 1 sinh ( 1 (44) If > 0 and < 1, then tanh = 1 cosh 1( 1+ 1 (45) If < 1, then tanh = 1 tanh ( 1+

4 386 fuguo ge and iquan liang and uzhong ding (46) If > 1, then coth = tanh ( 1 (47) If > 0 and 1, then sech 1 = cosh 1( 1 (48) If > 0 and 1, then sech = cosh ( 1 (49) If > 0, then csch = sinh ( 1 (50) If , then sinh +sinh = sinh ( (51) sinh sinh = sinh ( (5) If 1 and 1, then cosh 1 + cosh 1 = cosh 1( + ( 1) ( 1) (53) If 1 and 1, then cosh + cosh = cosh ( + ( 1) ( 1) (54) If 1 and 1 and, then cosh 1 cosh 1 = cosh 1( ( 1) ( 1) (55) If 1 and 1 and, then cosh cosh = cosh ( ( 1) ( 1) (56) If < 1 and < 1, then tanh + tanh = tanh ( + 1+ (57) If < 1 and < 1, then tanh tanh = tanh ( 1 (58) If 0 < and ( 1 +1 ) < 1, then log e = tanh ( 1 +1 (59) If 0 < and ( 1 +1 ) < 1, then log e = tanh ( 1 +1 (60) If 1 < and 1 +1, then log e = cosh 1( +1 (61) If 0 < and < 1 and 1 +1, then log e = cosh ( +1 (6) If 0 <, then log e = sinh ( 1 (63) If = 1 (ep ep( )), then = log e( (64) If = 1 (ep + ep( )) and 1, then = log e( + 1) or = log e ( + 1 ep ep( ) (65) If = ep +ep( ), then = 1 log e( 1+ 1 (66) If = (67) If = 1 ep +ep( ) ep ep( ) and 0, then = 1 log e( +1 1 ep +ep( ) (68) If = 1 ep ep( ), then = log e ( 1+ 1 ) or = log e ( 1+ 1 and 0, then = log e ( ) or = log e ( 1 1+ (69) (The function cosh)( ) = 1 + (the function sinh)(). (70) (The function cosh)() = 1 + (the function sinh)(). (71) (The function sinh)() = (the function cosh)() 1. (7) sinh(5 ) = 5 sinh + 0 (sinh) (sinh) 5.

5 formulas and identities of inverse (73) cosh(5 ) = (5 cosh 0 (cosh ) 3 ) + 16 (cosh ) 5. References [1] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91 96, [] Czes law Bliński. The comple numbers. Formalized Mathematics, 1(3): , [3] Jaros law Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1():69 7, [4] Rafa l Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5): , [5] Takashi Mitsuishi and Yuguang Yang. Properties of the trigonometric function. Formalized Mathematics, 8(1): , [6] Librar Committee of the Association of Mizar Users. Binar operations on numbers. To appear in Formalized Mathematics. [7] Konrad Raczkowski and Andrzej Nȩdzusiak. Real eponents and logarithms. Formalized Mathematics, ():13 16, [8] Andrzej Trbulec and Czes law Bliński. Some properties of real numbers. Formalized Mathematics, 1(3): , [9] Yuguang Yang and Yasunari Shidama. Trigonometric functions and eistence of circle ratio. Formalized Mathematics, 7():55 63, Received Ma 4, 005