On the Natural Logarithm Function and its Applications

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1 On the Natral Logarithm Fnction and its Applications By Edigles Gedes Febrary 4, 8 at March 7, 5 We love him, becase he rst loved s. - I John 4:9 Abstract. In present article, we create new integral representations for the natral logarithm fnction, the Eler-Mascheroni constant, the natral logarithm of Riemann eta fnction and the rst derivative of Riemann eta fnction.. Introction In this paper, we prove the amaing integral representations for the natral logarithm fnction, the Eler-Mascheroni constant, the natral logarithm of Riemann eta fnction and the rst derivative of Riemann eta fnction, as follows: ln ( ) ( ln )( ln ) ; ln sin sin( p ) p e p t ; and ( ) ln ( ) ( ln ) (ln )(ln ) ; ln + ln ( ln ) t e t ( + t) ; t ( + t)t ; sin F ; ; ( )() ln () (ln ) ln ( )() + ; ( )ln () t 6( + t)[ + ( + t)] 6 ( + t)t (3) 6( + t)[ + ( + t)] t[ ( + t) 6t(3)] 6 ( + t)t 3 :. Definition and Lemma Denition. The natral logarithm fnction can be dened by the Frllani integral as follows e ln t e t ; () t for Re() >. Lemma. If Re() >, then t e t : () ( + )

2 On the Natral Logarithm Fnction and its Applications Proof. We note that the left hand side of () can be written as t B(; ) () () ( + ) () ( + ) t e t ; (3) ( + ) 3. Theorems Theorem 3. If Im() / or Re(), then ln ( ) ( ln )( ln ) ; (4) where ln denotes the natral logarithm fnction. Proof. Sbstitting (3) in (), we enconter and Observe that ln e t t e t e t t e t t e t e t X k e t X k ( ) k t k t t t e t : (5) (6) ( ) k k t k : (7) We take (6) and (7) into (5) and obtain ln t X k ( ) k t k X t ( ) k k t k k X ( ) k k t t k X ( ) k k k t t k X ( ) k (k + )( ln ) (k+) X ( ) k k (k + )( ln ) (k+) k k X ( ) k ( ln ) (k+) X ( ) k k ( ln ) (k+) k k X k (ln ) (k+) X (ln ) (k+) k k X k (ln ) (k+) X (ln ) (k+) k k (ln ) (ln ) ( ) ( ln )( ln ) ; Theorem 4. If Re() >, then ( ln ) + ; (8)

3 3 where ln denotes the natral logarithm fnction. () e t t ; (9) for Re() >. We sbstitte (3) into (9), and nd () We take (6) into (), and enconter it implies that () X k () ( + ) e t t X k t e t t t e t t t : X k ( ) k 4.. The Natral Logarithm Fnction. Corollary 5. If Re() >, then ( ) ( ) k t k t t t +k t ( ) k ( ln ) (k++) (k + + ) X ( ) k( ln ) (k++) (k + + ) k ( + ) ( ln ) + ; ( ln ) +, ln ( ) 4. Applications where ln denotes the natral logarithm fnction. ( ln ) + ; () (ln )(ln ) ; () Proof. Dierentiating both sides in (4) with respect to, we obtain the desired reslt. 4.. The Eler-Mascheroni constant. Corollary 6. We have ( ln ) ln + ; ln ( ln ) where denotes the Eler-Mascheroni constant and ln denotes the natral logarithm fnction. + X k k k k d : ()

4 4 On the Natral Logarithm Fnction and its Applications From (8) and (), it follows that ( ln ) k+ k d ( ln ) + + X k k X + ( ln ) k+ X k k k d k ( ln ) + ( ln ) ln X ln k ( ln ) k+ ln( ln ) ( ln ) ln ln X ln( ln ) ( ln ) k+ k ( ln ) ln + ln ( ln ) ln( ln ) ( ln ) ln + ; ln ( ln ) 4.3. The Logarithm of the Riemann eta Fnction. Corollary 7. If Re() >, then ( )() ln () (ln ) ln ( )() + ; ( )ln where ln denotes the natral logarithm fnction and () denotes the Riemman eta fnction. Proof. We knew [, page 64] that ln () ln[( )()] + ln : We se the Theorem 3 and enconter ( )() ln () [( )() ln ]( ln ) + ln ( ln ) ( )() [( )() ln ]( ln ) + ( ) [ ( )ln ]( ln ) ( )() ( ln ) ( )() ln + ( )ln (ln ) ( )() ln ( )() + ( )ln ; 4.4. Again the Natral Logarithm Fnction. Corollary 8. If Re() >, then ( ) ln ( + t)( + t) ; (3) where ln denotes the natral logarithm fnction. Proof. We take e t in Theorem 3, and this completes the proof The Riemann eta Fnction.

5 5 Corollary 9. If Re(s) >, then (s)(s) (s ) Li s ( t) ; ( + t)t where (s) denotes the Riemann eta fnction and Li a (t) denotes the polylogarithm fnction. Proof. Changing of members and ln in (3), it follows that ln ( + t)( + t) : (4) Let e into (4) and obtain e (e + t)( + t) : (5) On the other hand, we well-know that (s)(s) s e ; (6) for Re(s) >. Sbstitting (5) into (6), we nd (s)(s) s (e + t)( + t) s + t e + t Li (s ) s ( t) ; ( + t)t therefore, we dece easily that (s)(s) (s ) Li s ( t) ; ( + t)t 4.6. Once Again the Natral Logarithm Fnction. Corollary. If Re() >, then ln sin sin( p ) p e p t ; (7) where ln denotes the natral logarithm fnction and sin denotes the sine fnction. Proof. In [], we nd (s + w )(s + a ) w a sin(a) sin(w) e s ; (8) a w for a / w. p p We get s t, w and a in (8) and obtain ( + t)( + t) sin sin( p ) p e p t : (9) Integrating (9) from at innity with respect to t, we enconter ( ) ( + t)( + t) sin sin( p ) p e p t : () We sbstitte the left hand side of (3) in the left hand side of (), and this conclde the desired proof.

6 6 On the Natral Logarithm Fnction and its Applications Qestion. Prove that, if Re() >, then ( ) ln ( + t)( + t) ; () where ln denotes the natral logarithm fnction Again the Eler-Mascheroni Constant. Corollary. We have t e t ( + t) ; t ( + t)t ; where denotes the Eler-Mascheroni constant, e denotes the eponential fnction and (a; ) denotes the pper incomplete gamma fnction. e v ln vdv: () We se () into () and have e v (v ) ( + vt)( + t) dv (v )e v dv + t + vt t e t ( + t) ; t ( + t)t ; Corollary 3. We have sin F ; where denotes the Eler-Mascheroni constant, sin denotes the sine fnction and F () denotes the Dawson's integral. Proof. We se (7) into () and have e v sin sin( p v ) p e p t dv v p sin( v )e v sin p dv e p t v h i sin F e p t e p h i t sin F h i sin F e p t sin F ; Qestion 4. Prove that lim e t (; t) ln ;

7 7 where denotes the Eler-Mascheroni constant, (a; t) denotes the pper incomplete gamma fnction and ln denotes the natral logarithm fnction. Corollary 5. If Re(s) >, then (s) + t X n s n ; (3) n + t n where (s) denotes the rst derivative of Riemann eta fnction. (s) X ln n n s ; (4) n for Re(s) >. We sbstitte (3) into (3) and enconter (s) X (n ) n s n (n + t)( + t) X + t n s n ; n + t Remark 6. If s and 3 in (3), we enconter the integral representations: () t 6( + t)[ + ( + t)] 6 ( + t)t ; (3) 6( + t)[ + ( + t)] t[ ( + t) 6t(3)] 6 ( + t)t 3 : n References [] Havil, Jlian, Gamma: Eploring Eler's Constant, First Edition, Princenton University Press, 3. [] available in March 8, 5.

This Topic follows on from Calculus Topics C1 - C3 to give further rules and applications of differentiation.

CALCULUS C Topic Overview C FURTHER DIFFERENTIATION This Topic follows on from Calcls Topics C - C to give frther rles applications of differentiation. Yo shold be familiar with Logarithms (Algebra Topic