In standard form x^2(d^2y/dx^2)+x(dy/dx)+(x^2-n^2)y=0 last term is (x^2-n^2). How to tackle the problem if instead of minus sign we have positive sign. i.e: (x^2+n^2)
Dear mohamed Ali,
the original form of Bessel differential equation is:
x2(d2y/dx2)+x(dy/dx)+(x2-n2)y=0
if you put n =i m ---> n2=-m2, thus Bessel differential equation transforms to the form:
x2(d2y/dx2)+x(dy/dx)+(x2+m2)y=0
which has the same solution, but with imaginary argument ( i m)
See the attached Mathematica Solution.
The following link also has information about the solution:
http://dlmf.nist.gov/10.24
Best Luck.
I guess either you are giving incorrect information about the problem or may be I understood it wrong. Standard form should be (x^2-n^2) or (x^2-n(n+1)). And as per the standard definition n can be complex and has the same result independent of the sign of n if n is real.
I think you should look for the original steps involved in obtaining the standard solution and from there you can get an idea respective to your problem.
Dear mohamed Ali,
the original form of Bessel differential equation is:
x2(d2y/dx2)+x(dy/dx)+(x2-n2)y=0
if you put n =i m ---> n2=-m2, thus Bessel differential equation transforms to the form:
x2(d2y/dx2)+x(dy/dx)+(x2+m2)y=0
which has the same solution, but with imaginary argument ( i m)
See the attached Mathematica Solution.
The following link also has information about the solution:
http://dlmf.nist.gov/10.24
Best Luck.
Dear Muhammad Ali.
I agree with Khaled Mohamed. He gave you a good answer: clear and concise.
They are many good books with this subject as well as
1. G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Mathematical Library, Edition 1995, Reprinted 1996.
2. F. Bowman, Introduction to Bessel Functions (Dover Books on Mathematics), Dover Publications Inc. New York, first pub. 1958.
Good luck.
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