TR/08/90 July 1990 An Alternative Development of
Basic Functions by
D. M. DREW
Department of Mathematics and Statistics Brunel University
Uxbridge UB8 3PH
BRUNEL UNIVERSITY
SEP 1991
LIBRARY
w9198819
-1-
An Alternative Development of Basic Functions
The concept of a limit presents considerable problems to many students, yet often the derivative is defined and limits taken with little thought given to the consequences by the instructor, let alone by the student. In this paper we investigate, in a simple problem, the consequence of not proceeding to the limit. Our example indicates how Calculus reduces the range of expressions that occur, yet masks important processes that are of
interest and should be returned to later. It reveals a little of the rich area of Mathematics, known as 'basic' functions, a topic which is so
frequently dismissed in a few mutterings about taking the limit. The price that has to be paid for taking the limit occurs when reversing the limit
process, then difficulties can arise.
'Basic' function arise when solving equations involving q-differences, the q-difference of the function f(x) being defined historically as
( ) ( ) ( ) ( q 1 ) x . x f qx x f
∆f −
≡ −
So for example ∆x n = [n]x n − 1 where the basic number [n] = . 1 q
1 q n
−
− Setting qx = x+h and letting q → 1 ( ) .
dx x df
∆f →
However, in studying the simple q-difference equation ( )
x 1 x 1
∆y = +
it became apparent that the results could be obtained more easily if a fundamental change was made. Considerable symmetry and simplicity is achieved if we define
( )
Q x Q 1
Q f x Qx f
∆f(x)
⎟⎟ ⎠
⎜⎜ ⎞
⎝
⎛ −
⎟⎟ ⎠
⎜⎜ ⎞
⎝
− ⎛
≡
To avoid confusion we will call this the Q-difference, it is closely
associated with the q-difference, and it has similer properties, in
-2- particular as
dx f(x) df 1
Q → ∆ → However, a number of the changes are significant. As before ∆ x n − [ n ] x n − 1 but now the basic number
Q . 1 Q
... 1 Q
Q Q Q 1
Q Q 1 ] n
[ n n 1 n 3 n 3 n 1
n
−
−
−
− + + + +
=
−
−
=
We develop the 'basic' functions that arise in solving the simple Q- difference equation ;
x 1 ) 1 x (
y = +
∆ in the limit as Q → 1 these functions become l n ( 1 + x ) . We will obtain an explicit expression for the periodic f u n c t i o n t h a t r e l a t e s t h e t w o f o r m s o f t h e s o l u t i o n . T h e r e s u l t i n g relation is both interesting and simple. Moreover the relation could be used to introduce a more advanced topic, elliptic functions.
The solutions of basic equations corresponding to many important differential equations have been investigated. A recent paper by H. Exton [3] examines a basic analogue of Hermite's equation, q-difference
equations have attracted the attention of many mathematicians and in particular the Rev. Frank H. Jackson [4]. A useful review paper is that by C. R. Adams [1] which in turn refers to N. E. Norlund's bibliography [5].
The Q-difference Operation.
An interesting extension of many of the fundamental functions that arise in Calculus called 'basic' functions was studied at the turn of the century, notably by Frank H. Jackson. Jackson's papers are listed in his obituary by T. W. Chaundy [2], 'Basic' functions strictly arise when solving equations involving the q-difference operator.
x ) 1 q (
) x ( f ) qx ( ) f x (
f −
= −
∆ .
However, we propose to extend the term basic function to include the
closely related solutions of equations involving our Q-difference operator
-3-
( ) ( )
. Q x Q 1
Q f x Qx f x
∆f
⎥ ⎦
⎢ ⎤
⎣
⎡ −
⎥ ⎦
⎢ ⎤
⎣
− ⎡
= (1)
On writing [n] = , Q Q 1
Q Q n 1 n
−
−
which equals n 1 n 3 n 3 n 1 Q
1 Q
Q 1
Q − + − + − + − + − for integer n and tends to n as Q → 1, we readily verify that
1 n n
X 1
n n
x [n]
x
∆ 1
x logQ 1 Q Q 1 2
∆ 1
[n]x
∆x
+
−
−
=
⎭ =
⎬ ⎫
⎩ ⎨
⎧ ⎥
⎦
⎢ ⎤
⎣
⎡ −
=
(2)
Now the differential equation, , y(0) 0 x
1 1 dx
dy =
= + has solution
( ) for x 1 3
x 2 x x x 1 n y
3
2 + − − − − <
−
= +
= l (3)
1 x for 2x
1 x nx 1
+ − 2 + − − − >
= l .
The corresponding Q-difference equation is
( ) 1 x x x for x 1
x 1 x 1
∆y = − + 2 − 3 + − − − <
= + (4.1)
1.
x x for
1 x
1 x
1 x 1
4 3
2 + − + − − − >
−
= (4.2)
The solution suggested by the expansion (4.1) is
( ) ≡ − + − + − − − ]
4 [
x ] 3 [
x ] 2 [ x x x
f 2 3 4
(5) which is convergent for |x| <
Q
1 when 0 <Q < l or |x| < Q when Q > 1.
While the solution suggested by the expansion (4.2) is
−
−
− +
− +
−
⎟⎟ +
⎠
⎜⎜ ⎞
⎝
⎛ − Q 2 3 4
[4]x 1 [3]x
1 [2]x
1 x x 1 Q log Q 1 2 1
⎥⎦ ⎤
⎢⎣ ⎡
⎭ +
⎬ ⎫
⎩ ⎨
⎧ −
= x
f 1 x Q log Q 1 2 1
Q
which is convergent for |x| > Q when Q < 1 or |x| >
Q
1 when Q > 1.
Thus the regions in which these two expansions converge overlap. It is the interrelation between them that we will investigate in this paper.
Solutions of ∆y(x) = x 1
1
+ can of course differ by more than a constant.
-4- Because ∆y(x) = 0 when
( ) ⎥
⎦
⎢ ⎤
⎣
= ⎡ Q y x Qx y Setting x = Q 2u
y [ Q 2 u + 1 ] [ = y Q 2 u − 1 ]
we see that the solutions may differ by a function periodic in u and with period 1.
Consideration of the value x = 1 leads us to write
( ) D ( x , Q .
x f 1 Q Q log Q 1 2 x 1
f ⎥ ⎦ X + +
⎢ ⎤
⎣
⎡ −
= ) (7)
This function D(x,Q) will turn out to be periodic and we will deduce an interesting consequence of this relation.
Iterative Solutions
The Q-difference equation ∆ f(x) = , f ( ) 0
x 1
1
+ = 0 (8)
i.e. ( )
x 1
x Q Q 1 Q
f x Qx
f ⎥ +
⎦
⎢ ⎤
⎣
⎡ −
⎥ =
⎦
⎢ ⎤
⎣
− ⎡
can be solved iteratively for both Q < 1 and Q > 1 . We will deal first with the case Q < 1.
( ) f ( ) Q x
xQ 1
xQ Q Q 1 x
f + 2
⎟⎟ +
⎠
⎜⎜ ⎞
⎝
⎛ −
−
=
= ∑
∞
=
+ +
⎥ +
⎦
⎢ ⎤
⎣
⎡ −
−
0 n
1 2n
1 2n
xQ 1
xQ Q
Q 1 (9)
which on expansion
[4] . x [3]
x [2]
x x
4 3
2 + − + − − −
−
= (10)
The radius of convergence R = Q − 1 results from the pole at
Q
X = − 1 in (9).
The relation (7) becomes
( )
∑
∑
∞
=
+
∞ +
=
+ +
+ +
⎭ ⎬
⎫
⎩ ⎨
⎧ −
⎭ −
⎬ ⎫
⎩ ⎨
⎧ − + =
⎭ ⎬
⎫
⎩ ⎨
⎧ −
−
0 n
1 2n
1 2n Q
0 n
1 2n
1 2n
Q x, Q D
x Q Q
Q 1 x Q log Q 1 2 xQ 1
1 xQ Q
Q 1
-5- Giving
(x) D x 2 log 1 xQ
1 xQ Q
x Q
Q Q 0
n
1 2n
1 2n
0 n
1 2n
1
2n = +
− +
+ ∑
∑ ∞
=
+
∞ +
=
+ +
(11)
( ) .
Q Q 1
Q) x D(x,
D where Q
−
=
This function D Q (x) is remarkably small for Q in the interval 2 1 <Q<1;
some computed values of its maximum value for real x are given in the table, these behave as 2αα απ where α =
nQ π l . Maximum Value of D Q (x)
u = .75, x = Q 1.5
2 D
Q Q (x)
.9 5.15 E-80
.8 2.15 E-37
.7 3.25 E-23
.6 4.06 E-16
.5 7.77 E-12
.4 6.04 E-09
.3 7.91 E-07
.2 3.68 E-05
.1 1.03 E-03
.05 5.77 E-03
.01 3.75 E-02
The error involved in omitting from (11) D Q ( X ) is thus quite small unless Q<.5.
Interesting approximations to logQx are obtained by taking a finite number of terms from the left hand side of (11).
Further it is readily shown that
( )
{ } Q { } 2u
1 u 2
Q Q D Q
D + =
So that when treated as a function of u, D Q ( ) x is a function periodic in u
with period 1. It can be expressed as a Fourier sine series
-6-
( ) ( ) .
nβ sinh
2nπn 1 sin
nQ Q 1
D
1 n 2u n
Q ∑
∞
=
−
= l (12)
where
nQ β π
2
= l . Noting that β is negative for Q < 1, this sum can be rearranged to give
[ ] ∑ ∞ ( )
= + +
=
0 r 2u
Q .
cos2πo β
1 2r cosh
sin2πi nQ
Q π
D l
( )
( ) ( )
∑ ∞
= + β + β
β +
+ π +
π
= π
0
r 2 r 1 2 2 r 1
1 r 2
e . u 2 cos e
2 1
u 2 sin e
2 Q
l n (13)
In the case Q > 1, equation (8) can be solved iteratively by writing
( ) ⎟⎟
⎠
⎜⎜ ⎞
⎝ + ⎛
⎟⎟ +
⎠
⎜⎜ ⎞
⎝
⎛ −
= 2
Q f x Q 1 x
Q x Q Q 1 x
f .
∑
∞
= +
+
⎟⎟ +
⎠
⎜⎜ ⎞
⎝
⎛ −
=
0
n 2n 1
1 2n
Q 1 X
Q X Q
Q 1 (14)
which on expansion
[ ] [ ] [ ] 2 x 3 x 4 .
x x
4 3
2 + − + − − −
−
=
The relation corresponding to (11) is
∑
∑ ∞
= +
∞
= + = +
− +
+ n 0 2 n 1 Q Q
0
n 2 n 1 log x D ( x )
2 1 1 xQ
1 x
Q
x (15)
and the effect is simply to replace Q by Q
1 , since
Q
D (x) = -D 1 Q (x).
In fact formula (12) for D Q [Q 2u ] still holds.
Integral Result
With x = Q 2u the relation (11) can clearly be integrated.
Consider the function ϕ
( ) ⎟
⎠
⎜ ⎞
⎝ ⎛ + +
π
=
ϕ ∞ + +
= x
Q 1 1 X Q
1 2 n 1 2 n 1
0 n
(16)
( 2n 1 2u )( 2n 1 2u
0 n
Q Q 1 Q Q 1
π + + −
∞
=
+ +
= )
-7-
[ ] ∑ ∞ ∑
=
∞
=
+
− + +
+
− +
= +
0
n n 0
2u 1 2n
2u 1 2n 2u
1 2n
2u 1 2n
Q 1 Q Q
Q 2 Q
Q Q 1
Q 2 Q
du log
d ϕ
Hence, with x = Q 2u , the relation (11) becomes [ ] ( )
) (Q 2D 2u
Q 2D Q
log du log
d
2u Q
2u Q 2u Q Q
+
=
+
=
− ϕ
and on integrating using the form (13) for D Q (Q 2u )
(17) consant
log u
log Q ϕ = 2 − Q θ +
−
where the function [ ( 2r 1 ) β 2 ( 2r 1 ) β ]
0 r
e cos2πo 2e
1 π
θ + +
∞
=
+ +
=
and .
nQ β π
2
= l
The significance of this result is clarified if we define q = e β ,
(18)
[ 2 r 1 4 r 2
0 r
q u 2 cos q 2
1 + +
∞
=
+ π +
π
=
θ ]
a Jacobi theta function . Then from (17), with l nq . l nQ = π 2
(19) AQ U
2ϕ .
= θ
An examination of the zeros of θ and of ϕ will disclose the nature of the relation between the two sides of this equation.
On putting u = 0, the constant A is given by
[ ]
[ 2 n 1 ] 2
0 n
1 2 r 2 0 r
Q 1
q 1 A
+
∞
=
+
∞
=
+ π
+ π
= (20)
when q = Q = e − π , A = 1.
Infinite Product form of Q u
2.
When Q < 1 the infinite products in (16) for ϕ are absolutely convergent and because l nq . l nQ = π 2 it follows q < 1.
( 1 q 2 r 1 e i 2 u )( 1 q 2 r 1 e i 2 u ,
0 r
π
− + π
+
∞
=
+ +
π
=
θ )
and again both infinite products in θ are absolutely convergent.
-8-
u
2Q can thus be expressed as the ratio of these infinite products
for and . By suitabl y changing t he vari able u, formula (19) provides θ ϕ a useful method of calculating all four of Jacobi's theta functions
especially for q near 1.
For Q > 1 we simply replace Q by Q
1 and q by q
1 and (19) becomes
. (21) ϕ
= θ AQ − u
2In particular when Q = e and q = e π
2( )
( ) ( ( ) )
( )
( 2r 1 π i2π2 ) ( ( 2r 1 ) π i2π2 )
0 r
2u 1 2n 2u
1 2n 0
u n
e e
1 e e
1 π
e e 1 e e 1 π A e
2 2
2
− +
− +
−
∞
=
+
−
− +
−
∞
=
+ +
+ +
= (22)
with
( )
( )
( )
( 1 e ) .
e 1 A
1 2 r 2 0
n
1 2 r 2 0
r
2
