() 2. x 2k+1- n. x <1

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1 Termwise Higher Derivative (Iv-Trigoometric, Iv-Hyperbolic). Termwise Higher Derivative of Iverse Trigoometric Fuctios.. Termwise Higher Derivative of arcta, arccot Formula.. Whe is ceillig fuctio, the followig epressios hold for <. () ( )! - cot - () - - k k ( )! - (.t) (.c) arcta is epaded to Tylor series as follows. ta ( )! < Differetiatig both sides of this with respect to repeatedly, we obtai the followig. () ( )! () - k ( - )! () 3 - k ( - )! () 4-3 k ( -3 )! Here, cosiderig the relatio betwee the derivative order ad the first term k 0 of k Such a relatio ca be epressed by k 0 (.t). Ad, usig a ceilig fuctio cot - - ta 0 : + <0 : - Therefore, we obtai (.c) immediately. Sum of Taylor series of the higher derivative of arcta Formula 9..7 i "9 Higher Derivative" was as follows. () ( )! / + r r C +-r +-r, it is as follows.. The we obtai - -

2 From (.t) ad this, the followig equatio follows for <. k ( )! - ( )! / + r r C k +-r +-r Ad givig to this without cosiderig the covergece coditio, we obtai the followig special value. k ( )! ( )! / r r C +-r (.t') Eample C 0! C! C3 - C3 0-3! C4 3- C4 0.. Termwise Higher Derivative of arcsi, arccos Formula.. Whe is ceillig fuctio, the followig epressios hold for <. si () ( - )!! - (.s) cos - () - ( - )!! - (.c) si ( - )!! ( - )!!( )! ( )!! ( ) ( )!! ( )! ( - )!! ( )! ( )! ( )!!( - )!! Differetiatig both sides of this with respect to repeatedly, we obtai (.s). ( The umber of the first term of is the same as it of arcta. ) (.c) is obtaied immediately from cos - / - si. Formula..' r - r ( r- )!!( 4 -r-3)!! 0 (.so) - -

3 r - r 4 -r-!! ( )!! ( r- )!! (.se) Formula 9.3.' i "9 Higher Derivative" was as follows. si ( ) - - From this, si si From (.s), si si k + 0 k Thus, we obtai the desired epressios. Eample ( r- )!!( -r-3)!! r ( +) r ( -) --r - r r - r r - r ( - )!! 0-0 ( - )! ( - )!! 0 - ( -)! ( r- )!!( 4 -r-3)!! ( r- )!!( 4 -r-)!! ( )!! 0 0 C3 0( -)!!5!!- C3!!3!!+ C3 3!!!!- C3 35!!( -)!! 0 C4 0( -)!!7!!- C4!!5!!+ C4 3!!3!!- C4 35!!!!+ C4 47!!( -)!! Termwise Higher Derivative of arccsc, arcsec Formula..3 The followig epressios hold for >. csc - () ( - )!! ( + )! --- ( )!! ( )! sec - () - ( - )!! ( + )! --- ( )!! ( )! (3.c) (3.s) csc - si ( - )!! ( )!!( ) -- > Differetiatig both sides of this with respect to repeatedly, we obtai (3.c). (3.s) is obtaied immediately from sec - / - csc

4 . Termwise Higher Derivative of Iverse Hyperbolic Fuctios.. Termwise Higher Derivative of arctah, arccoth Formula..t Whe is ceillig fuctio, the followig epressios hold for <. tah - () ( )! - (.t) arctah is epaded to Tylor series as follows. tah - ( )! < Differetiatig both sides of this with respect to repeatedly, we obtai the desired epressio. ( The umber of the first term of is the same as it of arcta. ) Formula..c The followig epressio holds for coth - () - () >. ( + )! --- ( )! (.c) arccoth is epaded to Tylor series as follows. coth - ( )! ( )! -- > Differetiatig both sides of this with respect to repeatedly, we obtai the desired epressio... Termwise Higher Derivative of arcsih, arccosh Formula..s Whe is ceillig fuctio, the followig epressio holds for <. sih - () k ( - )!! - (.s) arcsih is epaded to Tylor series for < as follows. sih - k ( - )!! ( )!! ( ) k k ( - )!!( )! ( )!! ( )! ( - )!! ( )! ( )! ( )!! ( - )!! Differetiatig both sides of this with respect to repeatedly, we obtai the desired epressio. ( The umber of the first term of is the same as it of arcta. ) - 4 -

5 Sum of Taylor series of the higher derivative of arcsi Formula 9.4. i "9 Higher Derivative" was as follows. sih - () - / r - -r From (.s) ad this, the followig equatio follows for <. k ( - )!! - - / r - -r ( r- )!!( -3-r)!! --r + + -r- ( r- )!!( -3-r)!! --r -r- Ad givig to this without cosiderig the covergece coditio, we obtai the followig special value. Eample k ( -)!! - / ( +- )!!! -!! -!!! - 3!! -! 3!! +! 3!! + 3! 3!! +! 5!! + 3! 5!! - 4! 5!! - 5! 5!! - 4! 7!! - 5! 7!! +- 6! 7!! +- 7! r - -r 4 7!! ! 8 9!! +- 7! 3 6 ( r -)!!( -3-r)!! -r- Formula..c The followig epressios hold for >. cosh - () - ( )! + - k ( +)! -- ( )!! (.c) arccosh is epaded to Tylor series for as follows. cosh - ( - )!! log - - ( - )!!( - )! log k ( )!! - - k ( )!!( )! ( - )! log - - k ( )!! ( )! ( )!!( -)!! Differetiatig both sides of this with respect to repeatedly, we obtai the desired epressio

6 ..3 Termwise Higher Derivative of arccsch, arcsech Formula..3 Whe is ceillig fuctio, the followig epressios hold for 0< <. csch - () ( )! - k ( - )!! - (3.c) k ( - )! sech - () ( )! ( - )!! - - (3.s) k ( - )! csch - log +k( -) k- log -k k log - log - k ( - )!! ( )!! ( - )!!( )! ( )!! ( )! k ( - )!! ( )! ( )!( )!!( -)!! Differetiatig both sides of this with respect to repeatedly, we obtai the followig. csch - () - - k ( - )!! - k ( - )! csch - ()! - k ( - )!! - k ( - )! csch - () 3! k ( - )!! -3 k ( -3 )! 4 csch - () 3! - k ( - )!! 4-4 k ( -4 )! Here, cosiderig the relatio betwee the derivative order ad the first term k 0 of k 0 Such a relatio ca be epressed by k 0 usig a ceilig fuctio (3.c). Net, sech - log +k log - log - k, it is as follows.. The we obtai ( - )!! ( - )!!( )! log ( )!! -k ( )!! ( )! ( - )!! ( )! ( )!( )!! ( -)!! Differetiatig both sides of this with respect to repeatedly, we obtai (3.s). ( The umber of the first term of is the same as it of arcta. ) - 6 -

7 Sum of Taylor series of the higher derivative of arccsch Formula 9.4. i "9 Higher Derivative" was as follows. csch - () - r A r r -+r - + where A r are coefficiets as follows. r- A A A 3A 3 A 3 A A 4 A 4 A 3 4 A A 5 A 5 A 3 5 A 4 5 A A 6 A 6 A 3 6 A 4 6 A 5 6 A From (3.s) ad this, the followig equatio follows for 0< <. k ( - )!! - k ( - )! ( )! + r r A r r- - + Ad givig to this without cosiderig the covergece coditio, we obtai the followig special value. k ( - )!! k ( - )! - ( )! + r r r- - r A r (3.c') Eample!! 3!! 5!! 7!! ! 43! 65! 87!!! 3!! 5!! 7!! ! + 4! 64! 86! 3!! 5!! 7!! 9!! ! + 4! 63! 85! 07! 3!! 5!! 7!! 9!! ! - 4 6! 84! 06! Reewal Alie's Mathematics K. Koo - 7 -

Propagation of Error with Single and Multiple Independent Variables

Propagation of Error with Single an Multiple Inepenent Variables Jack Merrin February 11, 017 1 Summary Often it is necessary to calculate the uncertainty of erive quantities. This proceure an convention