PREEXPONENTIAL AND PRETRIGONOMETRIC FUNCTIONS,Communications in Applied Analysis 14 (2010), no. 1, 99-116

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Abstract: This article develops pretrigonometric functions and prehyperbolic functions. Some properties of such functions are also developed. The theory developed will enrich the topic of special functions which are largely employed in applications.
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  Communications in Applied Analysis 14 (2010), no. 1, 99-116   PREEXPONENTIAL AND PRETRIGONOMETRIC FUNCTIONS Ramkrishna B. Khandeparkar 1 , Sadashiv Deo 2  And D. B. Dhaigude 3 1 Smt. Parvatibai Chowgule College Of Arts & Science, Margao – Goa 403.601, India 2 12, Precy Building, Mala, Panaji – Goa. 403.001, India 3 Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University Aurangabad 431004 , Maharashtra State, India T ramshila2000@yahoo.com T   Abstract:  This article develops pretrigonometric functions and prehyperbolic functions. Some properties of such functions are also developed. The theory developed will enrich the topic of special functions which are largely employed in applications. Keywords : Preexponential functions, pretrigonometric functions, prehyperbolic functions, extended pretrigonometric and prehyperbolic functions. 1. INTRODUCTION. The exponential, trigonometric and hyperbolic functions occupy significant space in mathematical literature. Generally these type of functions are classified under the title of “elementary functions”. It is known that such functions possess interesting properties such as continuity, differentiability, integrability, etc. Among these the trigonometric and hyperbolic functions possess periodic properties. Trigonometric and hyperbolic functions are in fact a part of the study of exponential functions. In fact, the entire study of such functions is the valuable contribution of the Euler’s number “e”. The various properties of the elementary functions are taught as a part of introductory course in mathematics since these play an important role in sequential course of mathematics. One of the question that is at the centre of the present article is ‘can we determine as set of functions ( to be called as prefunctions ) possessing a sequence {f  n ( t ) , t ∈  R} which approaches as n →  0 to one of the elementary functions’. The answer to this question is positive. The newly discussed set of prefunctions indeed possess some of the properties of elementary functions, while some properties such as periodicity are lost. In turn, the study here generalises the scope of elementary functions. The presently known elementary functions have natural generalisation which is classified under the title “extended elementary functions”. The study here also aims at obtaining the prefunction sets for the extended elementary functions and establishing some properties of such functions. The authors hope that the newly stated prefunctions – theory may equally get wide and meaningful applications in future.  ______________ Received June 1, 2009 1083-2564 $15.00 © Dynamic Publishers, Inc.   D.B.Dhaigude    100 KHANDEPARKAR, DEO AND DHAIGUDE   2. EULER GAMMA FUNCTION. The gamma function  can be thought of as the natural way to generalize the concept of the factorial to non-integer arguments. Leonhard Euler came up with a formula for such a generalization in 1729. At around the same time, James Stirling independently arrived at a different formula, but was unable to show that it always converged. In 1900, Charles Hermite showed that the formula given by Stirling does work, and that it defines the same function as Euler's. Euler's srcinal formula for the gamma function is ∏∏∏∏  ====++++∞∞∞∞→→→→ ++++++++====++++Γ ΓΓ Γ  n 0k 1z n )k 1z(  !n n Limit )1z( . However, it is now more commonly defined by ∫∫∫∫  ∞∞∞∞αααα−−−− ====++++ααααΓ ΓΓ Γ   0 t dt te )1(  ( 1 ) which converges for all αααα  > –1 . Using integration by parts, we get the important property of the Gamma function namely, Γ ΓΓ Γ   ( αααα  + 1 ) = αααα   Γ ΓΓ Γ   ( αααα  ) . ( 2 )   We let αααα  = 0 in ( 1 ) which results Γ ΓΓ Γ   ( 1 ) = 1 . For αααα  , a positive integer, Γ ΓΓ Γ   ( αααα  + 1 ) = αααα  ! . It is interesting to note that Γ ΓΓ Γ   ( αααα  ) is defined by ( 2 ) for all real values except αααα   = 0 , –1 , –2  , … Here   ( 1 ) does not give Γ ΓΓ Γ   ( αααα  + 1 ) for αααα   ≤  –1 because the behaviour of t P αααα P  at t = 0 makes the integral divergent. For details see [ 2 , p 45-46 ]  . The graph of Γ ΓΓ Γ   ( αααα  ) ( Figure 1 )   has the appearance as given below and is given to indicate the nature of Gamma function which we use in subsequent development of article. 3.   THE PREEXPONENTIAL FUNCTION OF A REAL VARIABLE. Exponential function play a wide role in almost all branches of mathematics. A question may be raised : are there a sets of functions a sequence from which tends to exp ( z ) ? We call such a set a prefunction set of the exponential function. Define most general form of exponential function in the form of series which we term as preexponential function. For any real number t and for any α   ≥  0 , α  being a parameter , define ∑∑∑∑ ∞∞∞∞====αααα++++αααα++++αααα++++αααα++++ αααα++++++++Γ ΓΓ Γ ++++====++++αααα++++Γ ΓΓ Γ ++++αααα++++Γ ΓΓ Γ ++++αααα++++Γ ΓΓ Γ ++++====αααα 1n n )1n ( t 1 ... )(t)(t)(t),t(pexp 4321 321 . ( 3 )    PREEXPONENTIAL AND PRETRIGONOMETRIC FUNCTIONS 101 Here pexp ( t , α  ) for each α   ≥  0 stands for prefunction of exp ( t ). Series ( 3 ) is absolutely convergent for all t ε  R. For each t ε  R and α   ≥  0 graphs of such prefunctions can be drawn. In particular when α  = 0 , ( 3 ) reduces to )t (exp )0 ,t (pexp  = .  In general , it is easy to observe that pexp (t , n) = pexp (t , n – 1) – !n t n  = pexp ( t , n – 2 ) – !)1n ( t 1n −−−− −−−−  = … , n = 1 , 2 , 3 , … Special cases of this general representation are now below :   t )t (exp ) ,t (pexp  −−−−==== 1 . ! t )t (exp ) ,t (pexp 22 2 t  −−−−−−−−==== . !3t !2t t )t (exp)3 ,t (pexp 32 −−−−−−−−−−−−====  = S )t (exp  3 −−−− where S B 3 B  is the partial sum of e P t – 1 .  In general, for n ∈  N, ∑∑∑∑ ==== ====−−−−==== n 1r rnn  !r t S whereS )t (exp)n ,t (pexp .  ( 4 ) Also, note that 1 )n ,t (pexp  n = ∞→  Limit   , for all t ε  R . It is interesting to compare the values of preexponential function at t = 0 , 1 , –1 for different integral values of n . We have Table 1 given below. Here the numbers in the second row keep on decreasing as n tends to ∞  and tend to 1 while those in the third row keep on oscillating. The graphical representation of the functions pexp ( t , 1 ) (Figure 2)   and pexp ( t , 2 ) (Figure 3) shown provide us the behavior of such functions. Note that curve (Figure 2)   is near parabolic in nature .  The curve lies in the first and second quadrant. In the next graph (Figure 3)   one can see the characteristic of Junction diode and Zener diode ( reverse bias ) .  Up to certain value the behavior is nonlinear and after it is like a linear function. Now, for general values of – t , 0 ≤  t < ∞  we have from ( 3 ) , ( )( ) ( ) . ...,3,2,1,0 ,  ) 1n( t1 1 1  ...) α 4( Γ t) α 3( Γ t) α 2( Γ t11),t (pexp 1nnn α 3 α 2 α 1 =αα++Γ  −−+=  −+++−+−−=α− ∑ ∞=α+α+++α  ( 5 ) Clearly from ( 5 )    102 KHANDEPARKAR, DEO AND DHAIGUDE   ,)t (exp ... )4( Γ t)3( Γ t)2( Γ t1)0 ,t (pexp 32 −−−−====−−−−++++−−−−====−−−−  We write as in ( 4 ) above , ∑ = −=−−=− n 1r rrnn  !r t )1( S whereS )t (exp)n ,t (pexp . Now we define a number e α  by ∑ ∞=α α++Γ  +=α= 1n )1n ( 1 1 ),1(pexp e  ; α   ≥  0 . and note that e e  ==== αααα→→→→αααα 0  Limit  .  Further , ∑ ∞=α−α α++Γ  −−+=α−= 1n n1 )1n ( )1( )1( 1 ),1(pexp e . }e e{  21 ee and ,}e e{  21 ee ,0 When 1111  −−αα−−αα  −=−+=+=α .  Further , observe that e α  > 1 and )( αααα++++Γ ΓΓ Γ  21  < 11 ====Γ ΓΓ Γ  )2( , )( αααα++++Γ ΓΓ Γ  31  < 211 ====Γ ΓΓ Γ  )3( , )( αααα++++Γ ΓΓ Γ  41  < 2 21 )4( <<<<====Γ ΓΓ Γ  611   , … 3 21 1 1 ... e  <<<<−−−−++++====++++++++++++++++++++<<<< αααα 121212111 32  , hence 3 e  <<<<<<<< αααα 1 . The problem of discovering additional properties of e α  is open. 4. PRETRIGONOMETRIC FUNCTIONS The prefunctions of !)n 2( t )1( t cos 0n n2n ∑∑∑∑ ∞∞∞∞==== −−−−==== and ∑∑∑∑ ∞∞∞∞====++++ ++++−−−−==== 0n 1n 2n !)1n 2( t )1( sin t are defined below :  ;R t ,  1n)  αα n2tn1)( 1  .)  αα 6t)  αα 4t)  αα 2t1 ) ,t (cos ) α  ,t ( 1X  1 n 2( Γ  ..7( Γ 5( Γ 3( Γ ∈ ∑ ∞= ++−+=+++−+++++−=α= +  p  ( 6 ) and    PREEXPONENTIAL AND PRETRIGONOMETRIC FUNCTIONS 103 .R t ,  0n)  αα  1 n 2 tn 1)( ... )  αα 5t)  αα 3t)  αα 1t ) ,t (sin )  α  ,t (  2X  2n 2( Γ  6( Γ 4( Γ 2( Γ ∈ ∑ ∞= +++−=−+++++−++=α= +  p  ( 7 ) Some particular cases for integral values of α  of these functions are listed below :  pcos ( t , 1 ) = sin t – t + 1, psin ( t , 1 ) = 1 – cos t, pcos ( t , 2 ) = – cos t – 2t 2  + 2, psin ( t , 2 ) = t – sin t, pcos ( t , 3 ) = – sin t + t – 6t 3  + 1, psin ( t , 3 ) = cos t + 2t 2  – 1, From these relations it is easy to obtain many trigonometrical relations .  For example pcos ( t 1  + t 2  , 1 ) = sin ( t 1  + t 2  ) – ( t 1  + t 2  ) + 1. Here the right hand side is completely known quantity. It is observed that the following formulae of psine and pcosine for α  = 2 n and α  = 2 n – 1 takes the form as follows : .....3,2,1n ,  !)1r 2( t )1()1( sin t )1( 1 )1n 2,t (pcos ;...3,2,1n ,  !)r 2( t )1()1( t cos)1( })1( 1{ )n 2 ,t (pcos n1r 1r 2r1n 1n n1r r2r1n n1n =−−−+−+=− =−−+−+−+= ∑∑ =−++=++   and . ...3,2,1n ,  !)r 2( t )1()1( t cos)1( )1( )1n 2,t (psin ,...3,2,1n ,  !)1r 2( t )1()1( sin t )1( )n 2 ,t (psin n1r r2rn 1n n n1r 1r 2rn n =−−+−+−=+ =−−−+−= ∑∑ =+=−   Note that psin ( t , 1 ) = 1 – cos t as mentioned above. Similarly for n ∈  N we state , . ...3,2,1n ,  !)1r 2( t )1()1( sin t )1( 1 )1n 2,t (pcos , )n 2,t (pcos )n 2 ,t (pcos n1r 1r 21r 1n n =−−−+−+=−− =− ∑ =−++ and   .)1n , t (psin )1n ,t (psin ...3,2,1n ,  !)1r 2( t )1()1( sin t )1( )n ,t (psin n1r 1r 21r n 1n −−−−====−−−−−−−−====−−−−−−−−−−−−++++−−−−====−−−−  ∑∑∑∑ ====−−−−++++++++ 222  
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