Integral Calculation: Analytical Calc of k_0 W L J_0

In summary, the conversation discusses the use of Mathematica to calculate the integral of a complicated function involving Bessel functions and sine integrals. The person asking the question is struggling with finding an analytical solution and suggests using numerical methods instead.
  • #1
kprokopi
2
0
I am trying to calculate (analytically) the integral:
[itex] \int_{0}^{\pi} \left [ \frac{ \sin( \frac{k_0 W \cos \theta}{2})}{\cos \theta} \right]^2 J_{0}(k_0 L \sin \theta ) \sin^3 (\theta ) d \theta [/itex]
where [itex] k_0, W, L [/itex] are constants and [itex] J_0 [/itex] is the Bessel function of the first kind of order zero.

Hint: Maybe we can use sine integrals [itex] Si(x)=\int_{0}^{x} \frac{\sin(\tau)}{\tau} d \tau [/itex].

Thanks in advance,
kprokopi
 
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  • #2
Eeh, you don't happen to be a masochist or something (:wink:)??

Use Mathematica and see what it spits out.
 
  • #3
That looks tricky.
Can't you cheat and use the trapezium rule with a large number of intervals? -_-;;
 
  • #4
Use Mathematica and see what it spits out.

Just try typing that monster into mathematica :rofl:
 

What is integral calculation?

Integral calculation is a mathematical process used to determine the area under a curve or the value of a function over a given interval. It involves breaking down a complex shape or function into smaller, simpler parts and using calculus principles to find the total area or value.

What is analytical calculation?

Analytical calculation is a method of solving mathematical problems using analytical techniques, such as algebra, trigonometry, and calculus. It involves using known formulas and equations to find exact solutions to problems, rather than using approximation methods.

What is k_0 in integral calculation?

k_0 is a constant that represents the starting point or initial value of the integral. It is usually used when finding the definite integral of a function over a specific interval, where k_0 would be the lower limit of integration.

How is W used in integral calculation?

In integral calculation, W represents the width or length of the interval over which the function is being integrated. It is typically used to determine the size of each small part that the function is broken down into in order to find the total area or value.

What is the purpose of J_0 in integral calculation?

J_0 is a variable that represents the function being integrated. It is often used to represent a specific function or to represent a generic function with a variable input. The purpose of J_0 is to help set up the integral equation and to evaluate the function at specific points.

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