Zero's of the modified Bessel functions,

In summary, the conversation discusses finding the solution to Airy's differential equation by fitting it to boundary conditions using a linear combination of Ai(x) and Bi(x) which involve modified Bessel functions of the first and second kind. The question asks about the zeros of these functions, and a paper is suggested as a potential resource for further information.
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thumper
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I have the solution to a particular D.E. (Airy's D.E.) which is in terms of Airy functions, namely a linear combination of Ai(x) and Bi(x), to which I have to fit to the boundary conditions. Both Ai(x) and Bi(x) can be cast into a form which involves both modified Bessel functions of the first and second kind. The bondary conditions are such that the solution must be zero at the boundary. Therefore my question is this,

What are the zero's of the modified Bessel functions of the first and second kind?
 
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1. What are the modified Bessel functions?

The modified Bessel functions are a family of special functions in mathematics that are used to solve differential equations and describe physical phenomena in various fields such as physics, engineering, and statistics. They are denoted by the symbols Iν(x) and Kν(x), where ν is a non-negative integer or half-integer and x is the argument of the function.

2. How are the modified Bessel functions related to the ordinary Bessel functions?

The modified Bessel functions are a generalization of the ordinary Bessel functions, which are denoted by Jν(x) and Yν(x). They differ from the ordinary Bessel functions in that they have a different form of the generating function and satisfy different differential equations. However, they are closely related and can be expressed in terms of each other using various identities and properties.

3. What are the properties of the modified Bessel functions?

The modified Bessel functions have many properties that make them useful in solving mathematical problems. Some of these properties include recurrence relations, integral representations, asymptotic behavior for large and small arguments, and special values at certain points. They also have connections to other special functions, such as hypergeometric functions and confluent hypergeometric functions.

4. How are the zeros of the modified Bessel functions related to their properties?

The zeros of the modified Bessel functions play an important role in their properties and applications. These zeros are closely related to the behavior of the functions, as they indicate the points where the functions are equal to zero or change sign. They also have applications in solving boundary value problems and finding eigenvalues in certain physical systems.

5. How can the zeros of the modified Bessel functions be calculated?

There are several methods for calculating the zeros of the modified Bessel functions, such as using numerical algorithms or special formulas. One commonly used method is the Steed's method, which involves finding the roots of a certain polynomial equation. Another method is to use asymptotic formulas and approximations to estimate the zeros. Additionally, tables and software programs are available to provide the values of the zeros for various arguments and orders of the modified Bessel functions.

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