Although we do not examine the properties of the Bessel functions in this book,
the related topics are dealt with in detail in the literature [5, 7–9]. The Bessel
functions are widely investigated as one of special functions in many branches of
mathematical physics.
To conclude this section, we have described an example to which the matrix A of
(15.179) that characterizes the system of differential equations is not a constant
matrix but depends on the parameter t. Unlike the case of the constant matrix A, it
may well be difficult to find a fundamental set of solutions. Once we can find the
fundamental set of solutions, however, it is always possible to construct the resolvent
matrix and solve the problem using it. In this respect, we have already encountered a
similar situation in Chap. 10 where the Green’s functions were constructed by a
fundamental set of solutions.
In this section, a fundamental set of solutions could be found and, as a consequence, we could construct the resolvent matrix. It is, however, not always the case
that we can recast the system of differential equations (15.66) as the form of
(15.179). Nonetheless, whenever we are successful in finding the fundamental set
of solutions, we can construct a resolvent matrix and, hence, solve the problems.
This is a generic and powerful tool.
References
1. Satake I-O (1975) Linear algebra (Pure and applied mathematics). Marcel Dekker, New York
2. Satake I (1974) Linear algebra. Shokabo, Tokyo. (in Japanese)
3. Rudin W (1987) Real and complex analysis, 3rd edn. McGraw-Hill, New York
4. Yamanouchi T, Sugiura M (1960) Introduction to continuous groups. Baifukan, Tokyo.
(in Japanese)
5. Dennery P, Krzywicki A (1996) Mathematics for physicists. Dover, New York
6. Inami T (1998) Ordinary differential equations. Iwanami, Tokyo. (in Japanese)
7. Riley KF, Hobson MP, Bence SJ (2006) Mathematical methods for physics and engineering, 3rd
edn. Cambridge University Press, Cambridge
8. Arfken GB, Weber HJ, Harris FE (2013) Mathematical methods for physicists, 7th edn. Academic Press, Waltham
9. Hassani S (2006) Mathematical physics. Springer, New York
References
617
the related topics are dealt with in detail in the literature [5, 7–9]. The Bessel
functions are widely investigated as one of special functions in many branches of
mathematical physics.
To conclude this section, we have described an example to which the matrix A of
(15.179) that characterizes the system of differential equations is not a constant
matrix but depends on the parameter t. Unlike the case of the constant matrix A, it
may well be difficult to find a fundamental set of solutions. Once we can find the
fundamental set of solutions, however, it is always possible to construct the resolvent
matrix and solve the problem using it. In this respect, we have already encountered a
similar situation in Chap. 10 where the Green’s functions were constructed by a
fundamental set of solutions.
In this section, a fundamental set of solutions could be found and, as a consequence, we could construct the resolvent matrix. It is, however, not always the case
that we can recast the system of differential equations (15.66) as the form of
(15.179). Nonetheless, whenever we are successful in finding the fundamental set
of solutions, we can construct a resolvent matrix and, hence, solve the problems.
This is a generic and powerful tool.
References
1. Satake I-O (1975) Linear algebra (Pure and applied mathematics). Marcel Dekker, New York
2. Satake I (1974) Linear algebra. Shokabo, Tokyo. (in Japanese)
3. Rudin W (1987) Real and complex analysis, 3rd edn. McGraw-Hill, New York
4. Yamanouchi T, Sugiura M (1960) Introduction to continuous groups. Baifukan, Tokyo.
(in Japanese)
5. Dennery P, Krzywicki A (1996) Mathematics for physicists. Dover, New York
6. Inami T (1998) Ordinary differential equations. Iwanami, Tokyo. (in Japanese)
7. Riley KF, Hobson MP, Bence SJ (2006) Mathematical methods for physics and engineering, 3rd
edn. Cambridge University Press, Cambridge
8. Arfken GB, Weber HJ, Harris FE (2013) Mathematical methods for physicists, 7th edn. Academic Press, Waltham
9. Hassani S (2006) Mathematical physics. Springer, New York
References
617