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# 10: Bessel Functions and Two-Dimensional Problems

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• 10.1: Temperature on a Disk
Let us now turn to a different two-dimensional problem. A circular disk is prepared in such a way that its initial temperature is radially symmetric,
• 10.2: Bessel’s Equation
Bessel's equation arises when finding separable solutions to Laplace's equation and the Helmholtz equation in cylindrical or spherical coordinates. Bessel functions are therefore especially important for many problems of wave propagation and static potentials.
• 10.3: Gamma Function
For ν not an integer the recursion relation for the Bessel function generates something very similar to factorials. These quantities are most easily expressed in something called a Gamma-function.
• 10.4: Bessel Functions of General Order
The recurrence relation for the Bessel function of general order ±ν can now be solved by using the gamma function.
• 10.5: Properties of Bessel functions
Bessel functions have many interesting properties.
• 10.6: Sturm-Liouville theory
In the end we shall want to write a solution to an equation as a series of Bessel functions. In order to do that we shall need to understand about orthogonality of Bessel function – just as sines and cosines were orthogonal. This is most easily done by developing a mathematical tool called Sturm-Liouville theory.
• 10.7: Our Initial Problem and Bessel Functions
We started the discussion from the problem of the temperature on a circular disk, solved in polar coordinates, Since the initial conditions do not depend on $$\phi$$, we expect the solution to be radially symmetric as well.
• 10.8: Fourier-Bessel Series
Fourier–Bessel series is a particular kind of generalized Fourier series based on Bessel functions and are used in the solution to partial differential equations, particularly in cylindrical coordinate systems.
• 10.9: Back to our Initial Problem