The Higher Derivatives Of The Inverse Tangent Function Revisited

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1 Alied Mathematics E-Notes, 0), 4 3 c ISSN Available free at mirror sites of htt:// The Higher Derivatives Of The Iverse Taget Fuctio Revisited Vito Lamret y Received 0 October 00 Abstract A closed-form formula for all derivatives of the real arctaget fuctio is reseted. I additio a curious series exasio for the fuctio is obtaied ad oe of its seci c cosequeces is give. Itroductio Costructig Maclauri series exasio for the arcta fuctio is easy by usig a itegral. However, Taylor series exasio aroud a arbitrary oit is ot so simle. It ca be easily veri ed, by iductio, that the fuctio arcta x ossesses o R derivatives of all orders. More recisely, there exists the sequece P of olyomials such that N d dx arcta x) P x) + x ) ad the degree of P x) does ot exceed. Obviously, these olyomials satisfy the recursio relatio P x) + x )P 0 x) xp x) with P 0 x). To our kowledge the closed form formula for P x) is still ukow. O the cotrary, may di eret ways of how to d cosecutive derivatives of arcta at x = 0 are kow, besides the method metioed above. Oe of them, for examle, is the iterative method. Namely, the fuctio yx) arcta x has the derivative y 0 x) +x ; cosequetly, the idetity + x )y 0 x) holds true. Hece, usig Leibiz rule for the -th derivative we obtai X + x ) k) y ) k) 0; k that is k=0 + x ) y +) x) + x y ) x) + ) y ) x) 0; ) for x R ad. Thus, we get the recursio EU y +) 0) = ) y ) 0); ; Mathematics Subject Classi catios: 65D0 y Faculty of Civil ad Geodetic Egieerig, Uiversity of Ljubljaa, Ljubljaa 000, Sloveia 386, 4

2 V. Lamret 5 which results i y k) 0) = 0 ad y k+) 0) = ) k k!); k 0: ) To geerate Taylor series exasio aroud a arbitrary x directly we eed the higher derivatives at this oit. However, to d y ) x) from ) it is ot easy. I [] the authors used a brilliat idea how to calculate it. Ufortuately, they were ot very careful i their aalysis ad made some errors i their derivatios ad i the al results as well. The fact that the Theorem i [] is ot valid is evidet from the observatio that the derivatives of arcta fuctio of eve orders are odd fuctios. However, the fuctios R, R x) beig the right had side of the Eq. ) i Theorem [], are eve for every. Figures a ad b, usig [3], show the grahs of the derivatives arcta 6) x) 40x 3 + 0x 3x 4 ) + x ) 6 ad arcta 8) x) 4030x 7x + 7x 4 x 6 ) + x ) 8, together with the grahs of the fuctios R 6 x) ad R 8 x) thick, dashed lies). We have the coicidece o R +, but ot o R Figure a: The grahs of the fuctios arcta 6) x) ad R 6 x) Figure b: The grahs of the fuctios arcta 8) x) ad R 8 x) Similarly, the sum, ad also all artial sums, of the series i [, Theorem, Eq. 6)] are eve fuctios, but arcta is a odd oe. Figure shows, usig [3], the grah of the fuctio arcta together with the grah of 500-th artial sum S 500 x) of the series o the right of Eq. 6) [] thick, dashed lie). The grahs coicide o R +, but evidetly ot o R Figure : The grahs of arctax) ad S 500 x) geerally: f odd eve) =) f 0 eve odd)

3 6 Higher Derivatives of the Iverse Taget Fuctio We wish to imrove the cotributio [] by givig the correct derivatios ad correct results. Higher Derivatives We shall show how the authors s idea ca be used successfully. To this e ect we reformulate the Theorem i its correct versio which di ers from the origial oe by the iclusio of factor sg x) where sgx) is de ed as, x < 0 sgx) :=, x 0. Hece sgx) is di eret from zero everywhere, x sgx) jxj, = sgx) sgx) ad sg x) = sgx) for x R r f0g. THEOREM. For x R ad there holds the equality arcsi d dx arcta x) = sg x) )! si + x ) = : 3) + x PROOF. The equality 3) is obviously true for = ad ay real x sice i this case the right-had side of the equatio 3) becomes equal to sg 0 x) 0! + x + x + x : Moreover, accordig to ), the relatio 3) is true also for x = 0 ad because the right-had side of the equatio 3) the becomes equal to sg 0) )! si ) )= )!, odd = 0, eve. Now we have to show that the idetity 3) is valid also for x R r f0g ad >. To do this we itroduce the auxiliary fuctio ': R! R, 'x) := arcsi 0; i ; 4) + x beig cotiuous ad di eretiable o R [ R + with the derivative ' 0 x) = Referrig to 4) we have x jxj + x = +x, x < 0 +x, x > 0. 5) si 'x) + x x R: 6)

4 V. Lamret 7 Cosequetly, there holds the equality ' 0 x) = si 'x), x < 0 *) si 'x), x > 0 **). 7) Remark. Cotrary to the suositio that was robably made by the authors [,. 7], the fuctio 'x) is ot di eretiable at x = 0. As a matter of fact, at this oit it has ite left ad right derivatives which are, ufortuately, di eret. Ideed, usig L Hôital rule we have ' 0 +) 0) = lim h"0 h#0) = lim h"0 h#0) arcsi h + h h jhj + h = + = ) : ) The grah of the fuctio 'x) is deicted, usig [3], i Figure Figure 3: The grah of the fuctio 'x) 0 Usig 4) ad 6), the equatio 3) is trasformed ito the followig equivalet idetity, for x R, d dx arcta x) = sg x) )! si 'x) si si 'x) : 8) For x R + the relatio 8) reduces to the equality d x arcta x) = ) )! si 'x) si si 'x) ; which, by iductio, could be easily veri ed [,. 7] usig 7). Hece 3) holds true for x > 0 ad. Cosequetly, for x R the relatio 5) is also valid sice i this case, substitutig x = t with t = tx) = jxj = x, we have d d d dt arcta x) = dx dx arcta tx) = arcta t) dt t= x dx x)

5 8 Higher Derivatives of the Iverse Taget Fuctio d = dt arcta t t= x = sg )! +x) + x ) = sg x) ) = si )! si + x ) = arcsi arcsi + x + x : ) 3 Curious Series Exasio The fuctio f give as comlex curviliear itegral, fz) := Z z 0 d + ; is aalytic o the cut comlex lae, i.e. i the domai D = C r fz C j Rez = 0; jimzj g ad there it has the comlex derivative f 0 z) = = + z ) [, Th. 3.5,. 8]. Particularly, we have f 0 x) = + x = arcta0 x); x R: Hece, fz) = arcta z for z R; f is a aalytic cotiuatio of real fuctio arcta. Due to its aalyticity, f ca be exaded ito Taylor s series aroud every z 0 D ad the obtaied ower series is coverget o every oe disk cetered at z 0 ad icluded i D [, Th. 6.7,. 36]. For ay x R the umber x + x) = 0 belogs to the oe disk jz xj < jx ij, which is icluded i D. Therefore, o this disk f ca be exaded ito Taylor s series; cosequetly 0 = f x + x) X f ) x) = fx) + x) ;! ad we get the exasio = arcta x = X = arcta ) x)! x) ; 9) true for every x R. Now, from 9) ad 3) we obtai the followig exasios arcta x = X = = sg x)! sg x) )! si + x ) = X x sg x) si + x ) = = arcsi arcsi + x + x x)

6 V. Lamret 9 = sgx) X j xj si + x ) = = Thus, we arrive at the followig theorem. arcsi THEOREM. For ay x R there holds the equality arcta x = sgx) X = : + x x = + x si arcsi : 0) + x I Figure 4 are deicted the grah of the fuctio arcta ad the grah dashed lie) of the 00-th artial sum of the series i the right had side of the equatio 0) Figure 4: The grah of arctax) ad its series aroximatio usig the 00-artial sum i 0) 4 -Series The immediate cosequece of Theorem is the followig result. THEOREM 3. For ' R such that 0 < j'j <, ad oly for such ', there holds the equality X j'j = sg') cos ') si '): ) = PROOF. A) 0 < ' < : I this case we cosider the variable x := q si ') > 0: We obtai ad ' = arcsi + x ad cot ' = x + x = + x = si ' = cos ' ) si ') = + x = x :

7 30 Higher Derivatives of the Iverse Taget Fuctio Cosequetly, sice ' 0;, it follows that ta ' = cot ' = x. Hece, ' = arcta x: 3) Uder give suositios, the relatios 3), 0) ad ) co rm the idetity ). B) < ' < 0 : Uder this coditio we estimate 0 < ' <. Cosequetly, cosiderig the recedig result, we have ') = X = cos ') si '); that is j'j = X = cos ') si '): Thus, the validity of the relatio ) is co rmed oce agai. C) < ' < : I this case the estimate 0 < ' + < usig the rst result, we obtai holds. Therefore, ' + ) = X = cos ' ) si '); that is j'j = X = cos ') si ') ad ) is aroved reeatedly. D) < ' < : Uder this coditio we have 0 < ' <. Thus, referrig to the rst result, we have ') = = X X = = cos ' cos ' si ') ) + si ') that is ' = X = cos ') si ') ad ) is veri ed also i this last case. The fuctio F, F ') := sg') P = cos ') si '), ful ll the idetities F '+ ) F '), for ' > 0, ad F ' ) F '), for ' < 0. Hece, the equality ) caot be true for ' R r [ ; ].

8 V. Lamret 3 Figure 5 illustrates the relatio ) by lottig, for ' 5 S k=0 3+k)+0:03; + k) 0:03, the grah of the fuctio ' 7! j'j dashed lie) ad the grah of the fuctio ' 7! sg') P 00 = cos ') si ') Figure 5: The grah of the fuctio ' 7! j'j dashed lie) ad its series aroximatio usig the 00-th artial sum Refereces [] K. Adegoke ad O. Layei, The higher derivatives of the iverse taget fuctio ad raidly coverget BBP-tye formulas for Pi, Al. Math. E-Notes, 000), [] A. I. Markushevich, Theory of Fuctios of a Comlex Variable, Vol., Pretice- Hall, Ic., Eglewood Cli s, N.J., 965. [3] S. Wolfram, Mathematica, versio 7.0, Wolfram Research, Ic.,