HP (Hewlett-Packard) 50g ユーザーズマニュアル

Page 16-55

The modified Bessel functions of the second kind,  

K

ν

(x) = (

π/2)⋅[I

-

ν

 (x)

−I

ν

 (x)]/sin 

νπ,

are also solutions of this ODE.

You can implement functions representing Bessel’s functions in the calculator in 
a similar manner to that used to define Bessel’s functions of the first kind, but 
keeping in mind that the infinite series in the calculator need to be translated 
into a finite series.

The functions T

n

(x) = cos(n

⋅cos

-1

 x),  and   U

n

(x) = sin[(n+1) cos

-1

 x]/(1-x

2

)

1/2

,

n = 0, 1, … are called Chebyshev or Tchebycheff polynomials of the first and 
second kind, respectively.  The polynomials Tn(x) are solutions of the differential 
equation (1-x

2

)

⋅(d

2

y/dx

2

)

− x⋅ (dy/dx) + n

2

⋅y = 0.

In the calculator the function TCHEBYCHEFF generates the Chebyshev or 
Tchebycheff polynomial of the first kind of order n, given a value of n > 0.   If 
the integer n is negative (n < 0), the function TCHEBYCHEFF generates a 
Tchebycheff polynomial of the second kind of order n whose definition is

U

n

(x) = sin(n

⋅arccos(x))/sin(arccos(x)).

You can access the function TCHEBYCHEFF through the command catalog 
(

‚N).

The first four Chebyshev or Tchebycheff polynomials of the first and second kind 
are obtained as follows:

 0 TCHEBYCHEFF, result: 1, 

i.e., 

T

0

(x) = 1.0.

-0 TCHEBYCHEFF, result: 1, 

i.e., 

U

0

(x) = 1.0.

 1 TCHEBYCHEFF, result: ‘X’,

i.e., 

T

1

(x) = x.

-1 TCHEBYCHEFF, result: 1,

i.e., 

U

1

(x) =1.0.

 2 TCHEBYCHEFF, result: ‘2*X^2-1, 

i.e., 

T

2

(x) =2x

2

-1.

-2 TCHEBYCHEFF, result: ‘2*X’, 

i.e., 

U

2

(x) =2x.

 3 TCHEBYCHEFF, result: ‘4*X^3-3*X’, i.e., 

T

3

(x) = 4x

3

-3x.

-3 TCHEBYCHEFF, result: ‘4*X^2-1’, 

i.e., 

U

3

(x) = 4x

2

-1.