- Email: [email protected]
Asymptotic expansion of an exponential function of fractional order
ASYMPTOTIC EXPANSION OF AN EXPONENTIAL FUNCTION OF FRACTIONAL ORDER* (ASIBPTOTICHESKOE FUNKTSII PAtAt
Vo1.25,
RAZLOZRENIE EKSPONENTSIAL'NOI DROBNOGO PORIADKA) No.4,
1961,
pp.
796-798
l3. D. ANN IN (Novosibirsk) (Received
The exponential has the form
function
April
25,
of fractional
1961)
order
introduced
?I=03 3nl” (Ifa) 3, (P. 1)= la n=O 2 r I@’ + 1)(I+ a)1
by Rabotnov
[ 1
1
(1)
(1 + a > 0)
We are interested in the asymptotic expansion for large t of this function and its derivatives with respect to t and p. and also of the integrals
this
The following article. Theorer
theorem,
which
I. Let us consider
was established
in [ 2 I,
is basic
for
the function
7X=03 E,
(29 q) =
2
r (7h,(;
4)
(z = z +
z”
iY> T > 0)
(2)
n=o where
4 is an arbitrary
constant,
Let h(n) be such that if w = z + iy in any half-space
l
real
or complex.
one considers z > z,,, then
the function
h(w),
where
While reading the proof sheets, the author became aware of the works of M.M. Dzhrbashian [ 4,5 1 in which were investigated. in particular, important properties of functions of the type 3, $3. t). 1192
Expansion
of
an
exponentihl
1193
function
h (w) I- bJ + 9) is a single-valued analytic modulus can be represented
function ID which in the form
. . . do not depend on ID, and where
cl,
large
values
of the
c, + 6 (W’, s) (P + 9) (P -!- 9 + I) I (-vu + 9 + s)
h(w)=C”+*-...where c,,, such that
for
lini 6 (Tu‘, s) = 0
for
the function
(3)
6(ytv,
s) is
1~’ 1--i K
Thus, for the function representations:
Er( z, 9) with large
when 0 < r < 2,
$ n
1 z 1, there
asymptotic
exist
(4)
1-Q -
-21
-2 esp
(z+)“~c,
z
(c,. c,....same as in@))
Y
(5)
when 0 < r < 2, 1arg z 1 = $n’i 1-9 -
I:‘, (2, 9) - +
--
2 -(
,I
,,=co
y-
2 r p9yn, 11 =*
Tl=O when r>,Z,
IargzI
i-”
(6)
?1 =m (7)
The first
it
summation is performed for all p for which
Let us restrict ourselves to the case 0 < a + 1 < 2. If ,8 > 0, then follows from (5), and if B < 0 from (4), that 1 3,
(p,
I)
-
13 l+a
a
exp (Ip’+t”)
*=03 1+a --n w ) 3, KL t) - - la z nx-2 rI(a+l)-(a-I-l)nl
(? > 0) (8) te < 0)
1194
If @ < 0 we deduce
from
(8)
B.D.
Annin
that
for
using Theorem 1. one AnalogouslY. pansions of a 3,@, t)/dt and a 3,@, tives with respect to t and 6 of the -3, cp. t). with
Furthermore,
The asymptotic Rozovskii [ 3 1. We note
that
the
aid
expansion
if
p is
of the
(10)
t
can show that the asymptotic will be equal to the t)/ap asymptotic expansion (8) for same theorem
with
a complex
large
one can
p < 0 was found
number,
one must
find
earlier
by
then
lim 3, (8, t) = 0 for + n (1 + a) < arg j3 6 (2 - WJ f-Ku lim 3, (8. 1) = 00 for iargBj<$n(a+l) l-co If a + 1 >, 2, then
exderiva-
use Formula
~6
(7).
BIBLIOGRAPHY 1.
Rabotnov, Iu. N., Ravnovesie uprugoi librium of an elastic medium with 1, 1948.
sredy s posledeistviem after-effect). PUM Vol.
(Equi12, No.
2.
Fry, C. G. and Hughes, H.K., Asymptotic developments of certain tegral functions. Duke Mathem. J. vol. 9. pp. 791-802, 1942.
in-
Expansion
3.
of
an
exponential
1195
function
Rozovskii. zuchesti krutki problem Izv.
M. I., Nelineinye integral’no-operatornye uravneniia poli zadacha o kruchenii tsilindra pri bol’shikh uglakh (Nonlinear integral operator equations of creep and the on the torsion of a cylinder with large torsion angles). Akad. Nauk SSSR, OTN, Mekhanika i rashinostroenie NO. 5, 1959.
4.
Dzhrbashian, hi. M., Ob integral’nom predstavlenii funktsii, nepreryvnykh na neskol’kikh luchakh (obobshchenie integrala Fur’ e) (On the integral representation of functions continuous on several rays (a generalization of the Fourier integral)). IZU. Akad. Nauk SSSR, seriia matem. 18, 1954.
5.
Dzhrbashian, primenenii formation tions).
M. M., Ob odnom novom integral’nom preobrazovanii v teoriiakh tselykh funktsii (On a new integral and its application in the theories of complete 12~. Akad. Nauk SSSR, seriia matem. 19, 1955.
Translated
i ego transfunc-
by
H.P.T.
Vo1.25,
RAZLOZRENIE EKSPONENTSIAL'NOI DROBNOGO PORIADKA) No.4,
1961,
pp.
796-798
l3. D. ANN IN (Novosibirsk) (Received
The exponential has the form
function
April
25,
of fractional
1961)
order
introduced
?I=03 3nl” (Ifa) 3, (P. 1)= la n=O 2 r I@’ + 1)(I+ a)1
by Rabotnov
[ 1
1
(1)
(1 + a > 0)
We are interested in the asymptotic expansion for large t of this function and its derivatives with respect to t and p. and also of the integrals
this
The following article. Theorer
theorem,
which
I. Let us consider
was established
in [ 2 I,
is basic
for
the function
7X=03 E,
(29 q) =
2
r (7h,(;
4)
(z = z +
z”
iY> T > 0)
(2)
n=o where
4 is an arbitrary
constant,
Let h(n) be such that if w = z + iy in any half-space
l
real
or complex.
one considers z > z,,, then
the function
h(w),
where
While reading the proof sheets, the author became aware of the works of M.M. Dzhrbashian [ 4,5 1 in which were investigated. in particular, important properties of functions of the type 3, $3. t). 1192
Expansion
of
an
exponentihl
1193
function
h (w) I- bJ + 9) is a single-valued analytic modulus can be represented
function ID which in the form
. . . do not depend on ID, and where
cl,
large
values
of the
c, + 6 (W’, s) (P + 9) (P -!- 9 + I) I (-vu + 9 + s)
h(w)=C”+*-...where c,,, such that
for
lini 6 (Tu‘, s) = 0
for
the function
(3)
6(ytv,
s) is
1~’ 1--i K
Thus, for the function representations:
Er( z, 9) with large
when 0 < r < 2,
$ n
1 z 1, there
asymptotic
exist
(4)
1-Q -
-21
-2 esp
(z+)“~c,
z
(c,. c,....same as in@))
Y
(5)
when 0 < r < 2, 1arg z 1 = $n’i 1-9 -
I:‘, (2, 9) - +
--
2 -(
,I
,,=co
y-
2 r p9yn, 11 =*
Tl=O when r>,Z,
IargzI
i-”
(6)
?1 =m (7)
The first
it
summation is performed for all p for which
Let us restrict ourselves to the case 0 < a + 1 < 2. If ,8 > 0, then follows from (5), and if B < 0 from (4), that 1 3,
(p,
I)
-
13 l+a
a
exp (Ip’+t”)
*=03 1+a --n w ) 3, KL t) - - la z nx-2 rI(a+l)-(a-I-l)nl
(? > 0) (8) te < 0)
1194
If @ < 0 we deduce
from
(8)
B.D.
Annin
that
for
using Theorem 1. one AnalogouslY. pansions of a 3,@, t)/dt and a 3,@, tives with respect to t and 6 of the -3, cp. t). with
Furthermore,
The asymptotic Rozovskii [ 3 1. We note
that
the
aid
expansion
if
p is
of the
(10)
t
can show that the asymptotic will be equal to the t)/ap asymptotic expansion (8) for same theorem
with
a complex
large
one can
p < 0 was found
number,
one must
find
earlier
by
then
lim 3, (8, t) = 0 for + n (1 + a) < arg j3 6 (2 - WJ f-Ku lim 3, (8. 1) = 00 for iargBj<$n(a+l) l-co If a + 1 >, 2, then
exderiva-
use Formula
~6
(7).
BIBLIOGRAPHY 1.
Rabotnov, Iu. N., Ravnovesie uprugoi librium of an elastic medium with 1, 1948.
sredy s posledeistviem after-effect). PUM Vol.
(Equi12, No.
2.
Fry, C. G. and Hughes, H.K., Asymptotic developments of certain tegral functions. Duke Mathem. J. vol. 9. pp. 791-802, 1942.
in-
Expansion
3.
of
an
exponential
1195
function
Rozovskii. zuchesti krutki problem Izv.
M. I., Nelineinye integral’no-operatornye uravneniia poli zadacha o kruchenii tsilindra pri bol’shikh uglakh (Nonlinear integral operator equations of creep and the on the torsion of a cylinder with large torsion angles). Akad. Nauk SSSR, OTN, Mekhanika i rashinostroenie NO. 5, 1959.
4.
Dzhrbashian, hi. M., Ob integral’nom predstavlenii funktsii, nepreryvnykh na neskol’kikh luchakh (obobshchenie integrala Fur’ e) (On the integral representation of functions continuous on several rays (a generalization of the Fourier integral)). IZU. Akad. Nauk SSSR, seriia matem. 18, 1954.
5.
Dzhrbashian, primenenii formation tions).
M. M., Ob odnom novom integral’nom preobrazovanii v teoriiakh tselykh funktsii (On a new integral and its application in the theories of complete 12~. Akad. Nauk SSSR, seriia matem. 19, 1955.
Translated
i ego transfunc-
by
H.P.T.