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Bessel Functions
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Bessel function of order
is a solution to the differential equation (Bessel equation)
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The Bessel functions are named after the German astronomer Friedrich Bessel (1784-1846). These functions first arose when Bessel solved Kepler's equation for describing planetary motion. These functions have been applied in many different physical situation, including the temperature distribution in a circular plate and the shape of a vibrating drumhead.
The Bessel function of order 0 is defined by
Let
.
Then
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=
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=
approaches to
for all
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Thus , by Ratio test , the series converges for all values of
. In other words, the domain of the Bessel function
is
.
Here are the first four partial sums and their graphs :
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We see that
, the Bessel function of order 0, is an even function with
. Moreover, by the Alternating Series Estimation Theorem, we have
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The animation below shows how the first 25 partial sums approach the Bessel function of order 0.
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The Bessel function of order 1 is defined by
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Let
.
Then
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=
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=
approaches to
for all
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Thus, by Ratio test, the series converges for all values of
. In other words, the domain of the Bessel function
is also
.
Here is the graph of
where
.
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We see that
, the Bessel function of order 1, is an odd function with
.
The animation below shows how the first 25 partial sums approach
.
Here we graph
and
on a common screen, can you see the relation between them ?
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It seems that
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, let's check.
=
=
=
To show that
satisfies the equation
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Let
, we have
=
=![]()
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=
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=![]()
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=
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=![]()
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Thus,
=
and
=
= 0
Therefore,
.
Similarly, we can show that
satisfies the equation
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For more information about the Bessel functions and the Bessel equations, please refer to
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http://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html
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