NASA TECHNICAL NOTE 2 NASA TN D-4296 -.-;K,- -

DEVELOPMENT OF A GENERAL FORMULA EXPANDING THE HIGHER-ORDER DERIVATIVES OF THE FUNCTION tanh 2 IN POWERS OF tanh 2 A N D A-NUMBERS

by Huns H. Hosenthien

George C. Murshall Space Flight Center Hzlntsvz’lle, Ala. - .

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION WASHINGTON, D. C. JANUARY 1968

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TECH LIBRARY KAFB, NM

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DEVELOPMENT OF A GENERAL FORMULA EXPANDING THE

HIGHER-ORDER DERIVATIVES O F THE FUNCTION

tanh z IN POWERS O F tanh z AND A-NUMBERS

By Hans H. Hosenthien

George C. Marsha l l Space Flight Center Huntsville, Ala.

N A T I O N A L A E R O N A U T I C S A N D S P A C E A D M I N I S T R A T I O N

For sale by the Clearinghouse for Federal Scientific and Technical Information Springfield, Virginia 22151 - CFSTl price $3.00

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TABLE OF CONTENTS

Page

SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

METHODOFDEVELOPMENT ............................ 1

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

L I S T OF TABLES

Page

9

10

. . . . . . . . . . . I.

11.

Derivatives of tanh z from first to sixth order

. . . . . . . . . . The A-numbers, Ar ( m ) , for r , m from 0 to 10

iii

DEVELOPMENT OF A GENERAL FORMULA EXPANDING THE HIGHERORDER DERIVATIVES OF THE FUNCTION

tanh z I N POWERS OF tanh z AND A-NUMBERS

SUMMARY

A general formula respect to z is developed.

fo r the rth derivative of the function tanh z with This formula is a finite polynominal in powers of

tanh z where the coefficients are of simple structure containing A-numbers.

The A-numbers, A ( m ) , of o r d e r m and degree r are introduced as an abbre-

viation for an expression containing C-numbers which are related to Euler 's numbers. A recursion formula for the A-numbers and their special properties are derived. The paper contains a tabulation of derivatives of tanh z from the first to the sixth o rde r as well as a table of A-numbers covering all combina- tions of o r d e r and degree from 0 to 10.

r

METHOD OF DEVELOPMENT

While deriving general t ransfer relations for electrical networks em- ploying synchronous commutation (modulation and demodulation) , it was found desirable to obtain closed-form expressions fo r the rth derivative of the function tanh z with respect to z.'"

Considering the Taylor series expansion of tanh ( z + y) in powers of y,

r we note that the r th derivative of tanh z is contained in the coefficient of y .

(tanh z) in powers of tanh z, we use a dr Since it is desired to express -

dzr

*The author is indebted to Prof. Dr. Richard F. Arenstorf of MSFC's Computa- tion Laboratory who suggested the basic approach as well as rigorous treat- ment of the infinite series involved.

I

second approach to develop tanh ( z + y) into a power series in y through the identity

tanh ( z + y) = (tanh z + tanh y) (I + tanh z . tanh y)-' (2 )

and the binomial expansion,

ob

( 3 ) m m (1 + tanh z . tanh y ) - '= 1 (-1) tanh z - tanhmy

m = O

m m which converges for I tanh z - tanh y I < 1. Combining equations ( 2 ) and ( 3) w e obtain

m m+ i

tanh ( z + y) = tanh z + 1 ( -l)m [tanh z - tanhm-'z] tanhmy m = l (4 1

( ( t a n h z l < i , l t anhyl < i ) . Now, for every fixed rea l value of z , the infinite s e r i e s on the right

of equation (4 ) is uniformly convergent in a closed c i rcu lar region of radius p around the origin of the y - plane defined by

For, abbreviating the general t e rms of the se r i e s by

f m ( y ) = [tanhm+'z - tanhm-'z] tanhmy

we have, fo r every fixed real z , I tanh z I < I and

Ifm(y) I < 2 ltanh y I m . To satisfy Weiers t rass 's M - test [ I] we require that

and find that this condition can be satisfied by the restriction

7r ly l 5 P <;I

2

7r which insures that any closed c i rcu lar region of radius p c - about the

or igin of the y - plane is mapped into a closed region in the interior of the unit c i rc le about the origin of the w - plane through the transformation

4

e'- i

e'+ i w = t a n h y = - .

m Now we substitute in equation (4) the power series for tanh y , developed below,

and obtain

lr 7r Since the s e r i e s (4) converges uniformly fo r I y I 5 p < 4 for every p < - and

since the series (5 ) converges at least f o r IyI < - , we can, by Weierstrass's

double-series theorem [ 11 , r eve r se the o rde r of summation over r and m in equation ( 6 ) and obtain the desired power series of tanh ( z + y) ,

4 7r

4

m+i tanh ( z + y ) = tanh z + C

Since both power series f o r tanh ( z + y ) , equations (I) and (7), have IT the region of convergence ly I < - in common and since their sums are equal

in this region, both power series are identical by the identity theorem fo r power series [ 11.

of equations ( 1) and (7) and, considering that A ( m ) = 0 for m > r, we obtain

4

Therefore we can equate corresponding coefficients

r

3

I

( z real, r > 0) . We note that the function tanh z , i ts derivative, and integral powers

are analytic functions for all complex z except fo r the poles of tanh z and that equation (8) holds along the real z - axis. Therefore, by the identity theorem for analytic functions [ I] , equation ( 8) holds for all complex z except fo r the poles of tanh z, and we obtain the desired expansion for the

r t h derivative of tanh z with respect to z in powers of tanh z:

The A-numbers, A (m) , are obtained from the recursion relation r

( r = O y i , . . . y ~ ; m = O , i y . . . y r ; m , r ~ 0)

where

=m! (m 2 0) ( m ) A m

A(m) = 0, when ( m - r ) is an odd integer in the range 1 5 m < r r

A ( m ) f 0, when ( m - r ) is an even integer in the range 1 < m 5 r.

Now we develop the power-series expansion of tanh y , stated in

r m

equation (5), which led to the introduction of the A-numbers, A ( m ) , with the r

recursion relation ( 10) and the properties ( 11) . Consider that the well known

4

power series of tanh y begins with y and contains only odd integral powers of y. Factoring y out of this series, we obtain tanh y in the form

t a n h y = y { l + % y 2 + a 4 y 4 + . . . } . ( 12)

Raising both sides of equation ( 12) to the mth power, w e have

( 13) m tanhmy=y { 1 + h ~ 2 + b ~ 4 . . . ) .

Note that the infinite series in brackets in equations (12) and (13) both contain only even powers of y , since the multinomial expansion of any finite o r d e r m y

{ I + +y2 + ay4 . . . lm, must again contain only even powers of y. equation (13) as

Writing

m m we see that the power series of tanh y starts with y powers of y when m is even and only odd powers of y when m is an odd integer.

and contains only even

To determine the coefficients of equation (14) , we start with the identity

2 tanh y = I - (e2'+ 1) '

Raising equation (15) to the mth power and using binomial expansion, we obtain

By the generating function for the C-numbers, C("), from Milne-Thomson [2 ] r

00

2n = E - tr ,(n) r r (et + I ) ~ r=O r ! 2

and after putting t = 2y, equation ( 17) becomes

where the C-numbers of o r d e r n and degree r are given by the recursion relation

5

The C-numbers are related to Euler’s numbers.

m Since, by equation ( i 4 ) , the lowest power in the series of tanh y is

ym and since, by equation (19), C(O) = 0 when r 2 I, the summation over

r mus t start with r = m and the summation over n must start with n = i i n equation ( 18). Thus, equation ( 18) becomes

r

m

t a d m y = r! r = m

Introducing the A-numbers , A(m) , by r

and substituting equation (21) into equation ( 2 0 ) , we have

which was stated in equation (5 ) .

Equation (21) expresses the A-numbers i n t e r m s of the C-numbers

where A:” is an A-number of order m and degree r. The A-numbers have the property stated in equation (11) which results from the discussion of equation ( 14).

The recursion relation ( I O ) for the A-number can be developed from equation ( 5) considering the properties stated in equation ( 11). Differentiat- ing equation ( 5) r-t imes and letting y = 0, we obtain

dr ( t adm r y = o

A(m) = [ dyr ’’1 6

I

Increasing m to m + I in equation (5) yields

co ( m + i ) E r' r '! A m+ I tanh y =

r' = m + l

Differentiating equation ( 23) (r + I) -times and letting y = 0 , we have

r+l

(m+i) - - [$ d( taah"y) l r+ I

y = o

dY y = o A

By equation ( 22) , equation (24) yields the desired recursion formula

which was stated in equation ( I O ) .

Now, to further evaluate equation ( 9 ) , we rewrite it as follows:

i A ( m ) z . m+ I

r = i A(m) tanh z - dr tanh z r dzr m = i m = l

A f t e r replacing m by m-2 in the first right hand summation and taking into -

( 0) = 0 and Ar = 0, when (rt-2) (r+l) = A r r account that, by equation ( 11) , A r > 0 , we obtain

7

Replacing m by m-2 in the recursion relation (IO) fo r A-numbers results in

Then substitution of equation ( 2 6 ) into equation ( 2 5 )

dz

yields

( 26)

(27)

Since, by equation (11) A(m) = 0 when ( m - r ) is an odd integer ( I 5 m < r)

we obtain the following formulas for m y r odd and m , r even, respectively: r

dz- m = 3

r > O ( 2 8 )

Derivatives of tanh z up t o the sixth o rde r are listed in Table I. The A-numbers,

A ( m ) , for r , m from 0 to 10 are listed in Table II. r

George C. Marshall Space Flight Center National Aeronautics and Space Administration

Huntsville, Alabama, August I O , 1967 933 -50-02 -0062

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TABLE I. DERIVATIVES OF tanh z FROM FIRST TO S E T H ORDER

d - (tanh z ) = i - tazh2z dz

d2 7 (tanh z) = - 2 tanh z + 2 tanh3z dz

d4 7 (tanh z) = 16 tanh z - 40 tanh3z + 24 tanh5z dz

d5 7 (tanh z) = 16 - 136 tanh2z + 240 tanh4z - 120 tanh6z dz

d6 -T (tanh z ) = - 272 tanh z + 1232 tanh3z - 1680 tanh5z + 720 tanh7z dz

9

TABLE 11. THE A-NUMBERS, A ! ~ ) , FOR r , m FROM o TO IO

2

0

0

2

0

- 16

0

272

0

-7936

0

353792

I

I i

m 3 4 5 6 7 8 9 10

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

6 0 0 0 0 0 0 0

0 24 0 0 0 0 0 0

-120 0 120 0 0 0 0 0

0 -960 0 720 0 0 0 0

3696 0 -8400 0 5040 0 0 0

0 48384 0 -80640 0 40320 0 0

-168960 0 645120 0 -846720 0 362880 0

0 -3256320 0 8951040 0 -9676800 0 3628800

P

0

I

0

-2

0

16

0

-272

0

7936

0

RE FER EN CE S

I. Knopp, Konrad: Theory of Functions, Part I, Elements of the General Theory of Analytic Functions. Dover Publications , New York, 1945, pp. 73, 81, 83, 85.

2. Milne-Thomson, L. M. , C. B. E. : The Calculus of Finite Differences. MacMillan & Co. , Ltd. , London, 1966, pp. 124-153.

NASA-Langley, 1968 - 13 M206

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