B 2k. E 2k x 2k-p : collateral. 2k ( 2k-n -1)!

Transcription

1 13 Termwise Super Derivative In this chapter, for the function whose super derivatives are difficult to be expressed with easy formulas, we differentiate the series expansion of these functions non integer times termwise. Therefore, e x, log x, sin x, cos x, sinh x, cosh x mentioned in " 1 Super Derivative " are not treated here Termwise Super Derivative of Trigonometric Functions & Hyperbolic Functions Formula Let ( x) be gamma function,, are ceilling function and floor function and let Bernoulli number B n and Euler number E k are as follows. B 0 =1, B =1/6, B 4 =-1/30, B 6 =1/4, B 8 =-1/30, E 0 =1, E =-1, E 4 =5, E 6 =-61, E 8 =1385, Then the following expressions hold for p 0 and 0< x < /. ( tan x ) p k k -1 B k k- p-1 x k ( k-p) ( tanhx ) p p +1 k= p +1 k= k k -1B k x k- p-1 k ( k-p) ( sec x ) p ( sech x ) p p +1 k= p +1 k= E k x k-p : collateral ( k-p +1) E k x k-p : collateral ( k-p +1) Proof There were the following formulas in " 10 Termwise Higher Derivative " Formula ( tan x ) Formula ( tanhx ) n +1 k= n +1 k= k k -1 B k k- n-1 x k ( k-n -1)! k k -1B k x k- n-1 k ( k-n -1)! Formula ( sec x ) n +1 k= E k x k-n ( k-n )! Formula ( sech x ) n +1 k= E k x k-n ( k-n )! In these formulas, replacing m! with gamma function ( 1+ m ) and analytically continuing the index of the differentiation operator to [ 0,p ] from [ 1,n ], we obtain the desired expressions. Since the termwise super integrals of arcsin x and arccos x were collateral, the super derivatives have to be collateral too. (This is the same in the following chapters.) - 1 -

2 Example 1: 3/4 th order derivative of tan x We calculated the the super differential coefficients on arbitrary one point x =0.4 according to the formula and Riemann-Liouville differintegral. As the result, two values were almost corresponding. Moreover, in the figure, blue shows tan x, red shows the 3/4 th order derivative and green shows the 1 st order derivative. Termwise super derivative of tan x tanp := (p,x)-> sum(^(*k)*(^(*k)-1)*abs(bernoulli(*k))/((*k *gamma(*k-p))*x^(*k-1-p),k=ceil((p+1)/)..00 (p, x) 00 k k 1 bernoulli( k) k 1 p x ( k) (k p) k p 1 Riemann-Liouville differintegral p:=3/4: h:=10^-10: f := x-> 1/gamma(1-p)*int((x-t)^(1-p-1)*tan(t), t=0..x) x 1 x (x t) 1 p 1 tan(t) d t (1 p) 0 Example : 9/10 th order derivative of tanh x Only the figure is shown. Blue shows tanh x, red shows the 9/10 th order derivative and green shows the 1 st order derivative. - -

3 Example 3: 1/ th order derivative of sec x We calculated the the super differential coefficients on arbitrary one point x =0.3 according to the formula and Riemann-Liouville differintegral. As the result, two values were almost corresponding. Moreover, in the figure, blue shows sec x, red shows the 1/ th order derivative and green shows the 1 st order derivative. Formula Let ( x) be gamma function,, are ceilling function and floor function and let Bernoulli number B n and Euler number E k are as follows. B 0 =1, B =1/6, B 4 =-1/30, B 6 =1/4, B 8 =-1/30, E 0 =1, E =-1, E 4 =5, E 6 =-61, E 8 =1385, Then the following expressions hold for p 0 and / < x <. ( cot x ) p k k -1 = - Σ x- k ( k-p) ( csc x ) p p +1 k= p +1 k= B k k- p-1 E k ( k-p +1) x- k-p : collateral - 3 -

4 Proof There were the following formulas in " 10 Termwise Higher Derivative " Formula ' ( cot x ) Formula ' ( csc x ) = - Σ n +1 k= n +1 k= k k -1 k( k-n -1)! B k x- E k ( k-n )! x- k-n k- n-1 In these formulas, replacing m! with gamma function ( 1+ m) and analytically continuing the index o the differentiation operator to [ 0,p ] from [ 1,n ], we obtain the desired expressions. Example 1: 3/4 th order derivative of cot x We calculated the the super differential coefficients on arbitrary one point x =1.7 according to the formula and Riemann-Liouville differintegral. As the result, two values were almost corresponding. Moreover, in the figure, blue shows cot x, red shows the 3/4 th order derivative and green shows the 1 st order derivative. Termwise super de rivative of cot x cotp := (p,x)-> -sum(^(*k)*(^(*k)-1)*abs(bernoulli(*k))/((* *gamma(*k-p))*(x-pi/)^(*k-1- p),k=ceil((p+1)/)..00) 00 (p, x) k k 1 bernoulli( k) x ( k) (k p) k 1 p k p 1 Riemann-Liouville differintegral p:=3/4: h := 10^-11: f := x-> 1/gamma(1-p)*int((x-t)^(1-p-1)*cot(t), t=pi/..x) x 1 x (x t) 1 p 1 cot(t) d t (1 p) - 4 -

5 Example : 14/15 th order derivative of csc x Only the figure is shown. Blue shows csc x, red shows the 14/15th order derivative and green shows the 1 st order derivative. The following lineal termwise super derivatives exist for csch x and sech x. Formula The following expressions hold for p 0, x >0. ( csch x ) p = -p ( k+1) Σ p k=0 Proof e ( k +1 ) x ( sech x ) p = -p Σ k k=0 Formula ( 8.1 ) was as follows. x x csch x dx p = p Σ k=0 ( k+1) p e ( k +1 ) x e -( k +1 ) x ( k+1) p x x sech x dx p = p Σ k k=0 e -( k +1 ) x ( k+1) p Since differentiation is the reverse operation of integration, replacing the index p of the integration operator with -p, we obtain the desired expressions. Example: 7/9 th order derivative of csch x. We calculated the the super differential coefficients on arbitrary one point x =3.8 according to the formula and Riemann-Liouville differintegral. As the result, two values were almost corresponding

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7 13. Termwise Super Derivative of Inverse Trigonometric Functions Formula When ( x) is gamma function and is ceilling function, the following expressons hold for p 0 and 0< x <1. tan -1 x p k= cot -1 x p x -p = ( 1-p) sin -1 x p k= k ( k )! ( k+-p) - Σ k= x k+1- p k ( k )! ( k+-p) x k+1- p ( k-1 )!! x k+1-p : collateral ( k+-p) cos -1 x p x -p = ( 1-p) - Σ k= ( k-1 )!! x k+1-p : collateral ( k+-p) Proof There were the following formulas in " 11 Termwise Higher Derivative ". Formula tan -1 x Formula sin -1 x n -1 k= n -1 k= k ( k )! ( k+1-n )! ( k-1 )!! x k+1-n ( k+1-n )! x k+1- n In these formulas, replacing m! with gamma function ( 1+ m) and analytically continuing the index of the differentiation operator to [ 0,p ] from [ 1,n ], we obtain tan -1 x p, sin -1 x cot -1 x = x 0 - tan -1 x p. Next, From this, cot -1 x p = x 0 p - tan -1 x p Substituting x 0 p x -p =, tan -1 x ( 1-p) p k= k ( k )! ( k+-p) for this, we obtain cot -1 x p. cos -1 x p is also obtained in a similar way. x k+1- p Note When p =1,,3,, ( 1-p ) =( 0 ),,(-), i.e. ( 1-p ) =. Then, x -p = 0 for p =1,,3, ( 1-p) Therefore, If we replace p with n, cot -1 x p, cos -1 x p results in the following formulas in

8 Formula cot -1 x Formula cos -1 x n = - Σ n -1 k= n = - Σ n -1 k= k ( k )! ( k+1-n )! ( k-1 )!! x k+1-n ( k+1-n )! x k+1- n Example 1: 9/10 th order derivative of arctan x We calculated the the super differential coefficients on arbitrary one point x =0.1 according to the formula and Riemann-Liouville differintegral. As the result, two values were almost corresponding. Moreover, in the figure, blue shows arctan x, red shows the 9/10 th order derivative and green shows the 1 st order derivative. Example : 1/ th order derivative of arccot x We calculated the the super differential coefficients on arbitrary one point x =0.05 according to the formula and Riemann-Liouville differintegral. As the result, two values were almost corresponding

9 Example 3: 4/5 th order derivative of arcsin x Only the figure is shown. Blue shows arcsin x, red shows the 4/5 th order derivative and green shows the 1 st order derivative. Example 4: 1st order derivative of arccos x When p =1, ( 1-p ) =( 0= ). Then cos -1 x 1 = -Σ k=0 x -p ( 1-p) ( k-1 )!! x k = - ( k+1) In fact, this series converges to the right-hand side on x <1. = 0. Therefore, from the formula, 1 1-x - 9 -

10 13.3 Termwise Super Derivative of Inverse Hyperbolic Functions Formula When ( x ), ( x ),, are gamma function, digamma function, ceilling function and Euler-Mascheroni constant (= ), the following expressons hold for p 0 and 0< x <1. tanh -1 x p ( k )! x k+1- p Proof k= sinh -1 x p k= sech -1 x p = ( 1-p) csch -1 x p = ( 1-p) ( k+-p) k ( k-1 )!! x k+1- p : collateral ( k+-p) x -p log +( ) x 1-p + -Σ k= x -p log +( ) x 1-p + - Σ k= p There were the following formulas in " 11 Termwise Higher Derivative " Formula 11..1t tanh -1 x Formula 11..s sinh -1 x n -1 k= n -1 k= ( k )! ( k+1-n )! p ( k-1 )!! x k-p k( k-p+1) : collateral k ( k -1)!! x k-p k ( k -p+1) x k+1- n k ( k-1 )!! x k+1-n ( k+1-n )! : collateral In these formulas, replacing m! with gamma function ( 1+ m) and analytically continuing the index of the differentiation operator to [ 0,p ] from [ 1,n ], we obtain tanh -1 x p, sinh -1 x Next, arcsech x is expanded to Tylor series for 0< x <1 as follows. sech -1 x = log x 0 - log x -Σ k=1 Differentiating both sides of this with respect to x n times, sech -1 x Substituting for this, ( k-1 )!! x k k ( k )! = log x 0 - ( log x ) x 0 x -n = ( 1-n, ( log x ) ) sech -1 x = ( 1-n) Replacing n with p, we obtain sech -1 x n = n - Σ n k= p. ( k-1 )!! x k-n k ( k-n )! log x -( 1-n) - x -n ( 1-n) x -n log +( ) x 1-n + - Σ k= n ( k-1 )!! x k-n k ( k-n )! p. csch -1 x p is also obtained in a similar way

11 Note According to Formula ( 1.3 ), ( 1-n) ( 1-n) = n ( n -1)! n =1,,3, If we substitute this for sech -1 x in the proof, it results in Formula ( 11. ) as follows. sech -1 x = n ( n -1)! ( k-1 )!! - Σ x n x k-n n k= k ( k-n )! Example 1: 3/4 th order derivative of arctanh x We calculated the the super differential coefficients on arbitrary one point x =0. according to the formula and Riemann-Liouville differintegral. As the result, two values were almost corresponding. Moreover, in the figure, blue shows arctanh x, red shows the 3/4 th order derivative and green shows the 1 st order derivative. Example : 3/ th order derivative of arcsinh x Replacing the calculation order of differentiation and integration in Riemann-Liouville differintegral, we obtain f p 1 x = ( n -p) x ( x-t) a n-p-1 d n dt n f t dt n= p We calculated the the super differential coefficients on arbitrary one point x =0.3 according to the formula and this expression. As the result, two values were corresponding

12 Example 3: 6/7 th order derivative of arcsech x Only the figure is shown. Blue shows arcsech x, red shows the 6/7 th order derivative and green shows the 1 st order derivative Renewal Alien's Mathematics K. Kono - 1 -