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With these definitions, a general solution of Bessel’s equation for all values of
ν
is given by y(x) = K
1
⋅
J
ν
(x)+K
2
⋅
Y
ν
(x).
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
H
n
(1)
(x) = J
ν
(x)+i
⋅
Y
ν
(x), and H
n
(2)
(x) = J
ν
(x)
−
i
⋅
Y
ν
(x),
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
ν
(x)= i
-
ν
⋅
J
ν
(i
⋅
x), where i is
the unit imaginary number. These functions are solutions to the differential
equation x
2
⋅
(d
2
y/dx
2
) + x
⋅
(dy/dx)- (x
2
+
ν
2
)
⋅
y = 0.
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.
Chebyshev or Tchebycheff polynomials
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
