Inside the spheres, they can be any smooth continuation, such as a linear combination of polynomials or Bessel functions.

42.

The Bessel function of the second kind then can be thought to naturally appear as the imaginary part of the Hankel functions.

43.

The Bessel functions are valid even for complex arguments, and an important special case is that of a purely imaginary argument.

44.

For instance, the Sonine formula is a formula given by Sonin for the integral of the product of three Bessel functions.

45.

The coefficients can be obtained from projecting the function " f ( x ) " onto the respective Bessel functions:

46.

If is the modified, or hyperbolic, Bessel function of the first kind, we may define a function on the rationals by

47.

The most familiar transcendental functions are the logarithm, the Bessel functions are transcendental in general, but algebraic for some special parameter values.

48.

For example, the solution for the time-dependent " start-up " process can be expressed in terms of Bessel functions.

49.

So all functions that are essentially special cases of it, such as Bessel functions, can be expressed as limits of hypergeometric functions.

50.

Dougall's own contributions to mathematics include works on Bessel functions, Mathieu functions, hypergeometric series, and the Schl�fli double six.

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