# Chebyshev Or Tchebycheff Polynomials, - HP 48gII User Manual

Graphing calculator.

With these definitions, a general solution of Bessel's equation for all values of
ν is given by
In some instances, it is necessary to provide complex solutions to Bessel's
equations by defining the Bessel functions of the third kind of order ν as
(1)
H
(x) = J
n
These functions are also known as the first and second Hankel functions of
order ν.
In some applications you may also have to utilize the so-called modified
Bessel functions of the first kind of order ν defined as I
the unit imaginary number. These functions are solutions to the differential
2
equation
x
The modified Bessel functions of the second kind,
K
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.
Chebyshev or Tchebycheff polynomials
The functions T
(x) = cos(n⋅cos
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
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
⋅J
y(x) = K
(x)+K
ν
1
(2)
(x)+i⋅Y
(x), and H
(x) = J
ν
ν
n
⋅(d
2
2
y/dx
) + x⋅ (dy/dx)- (x
(x)]/sin νπ,
(x) = (π/2)⋅[I
(x)−I
ν
ν
ν
-
-1
x), and U
(x) = sin[(n+1) cos
n
2
2
2
) − x⋅ (dy/dx) + n
)⋅(d
y/dx
⋅Y
(x).
ν
2
(x)−i⋅Y
(x),
ν
ν
-
ν
(x)= i
J
(i
x), where i is
ν
ν
2
2
) ⋅y = 0.
-1
x]/(1-x
2
⋅y = 0.
Page 16-57
2
1/2
)
, n