Use the appropriate series of the Bessel functions Jn(x) to compute the following values using t

Use the appropriate series of the Bessel functions Jn(x) to...

Key Concepts

Numerical Approximation and Series Truncation

In numerical computations, approximating an infinite series by using only a finite number of terms is common. Using the first four terms of the Bessel function series provides an approximation whose accuracy depends on the value of x and the order of the Bessel function. Understanding the error and convergence behavior of such series truncations is essential for ensuring reliable numerical results.

Gamma Function

The Gamma function extends the factorial function to non-integer values and is a crucial component of the series expansion for Bessel functions. It is especially important when dealing with Bessel functions of fractional or complex order, as it appears in the denominator of the series terms, ensuring that the series is properly defined for non-integral orders.

Bessel Functions

Bessel functions are solutions to Bessel's differential equation and play a key role in modeling physical phenomena with cylindrical symmetry, such as heat conduction, wave propagation, and static potentials. They are defined for various orders—including integer, fractional, and even complex orders—which affects the form of their series expansion and their properties.

Series Expansion of Bessel Functions

The Bessel functions can be represented as an infinite series. For example, the Bessel function of order n is given by the series J?(x) = ????? [(-1)?/(m! ?(m+n+1))]*(x/2)^(2m+n). This series representation underlies numerical methods for approximating these functions by truncating the series after a few terms, as required in practical computations.

Transcript

00:01 Okay, we have to first prove that the series converges for all real numbers.

00:05 So first, let's use the ratio test.

00:12 So we have to calculate the limit when a k plus 1 over a k of a k plus 1 over a k when k goes to infinity.

00:22 And that's the expression that we have.

00:24 Now we can make that division so we can simplify.

00:27 And we have k plus 1 on top and this one goes with that one we have x to the 2k plus 3 and x to the 2k over here so we have 3 left on top we have 2 to the 2k plus 1 and 2 to 2 k plus 3 on the bottom so we have 2 left over there we have k factorial and k plus 1 factorial and we have k plus 1 factorial and k plus 2 factorial.

01:06 So what we have left is that expression.

01:13 Now when k goes to infinity, this goes to 0.

01:18 And therefore we have that the zero converges for all x.

01:23 Now in the next point, what we have to do is check that this power series satisfy the differential equation that we have.

01:34 Now, if we take the derivative of j1, that would be this and there is no doubt about it, i think.

01:44 The index starts from zero because this series starts from 1.

01:50 So the next, when we take the derivative, we still start from 0.

01:55 Now, when we take the second derivative, we have to move it to 1 because we have to delete the term that here was degree 0.

02:03 So this is the expression that we have for the second derivative.

02:06 Now, the differential equation that we have to check is that one.

02:15 So we can multiply j double prime by, this should be j1, j1 double prime by x squared, that will make 2 more x over here.

02:37 So we used to have 2k, x2, x2.

02:41 The 2k, now we have to x to the 2k plus 2.

02:45 Now we can multiply by x j prime.

02:50 And so that gives us that the x is going to be 2k plus 1 this time.

02:57 Now for the last term, we have j1 multiply by x square, that's this term, and x1 multiply by negative 1.

03:09 That's that term.

03:11 So the next thing we are going to do is take the first term of the series and the first term of this series out.

03:19 So now we have the series starting from one for both of them.

03:24 And we take out a term that's one half and negative one half.

03:29 So these two can cancel out.

03:36 And now we have that expression over there.

03:38 Now we have three of them.

03:40 They're starting in one.

03:42 This one is starting at 0...

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