- #31
Mandelbroth
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Your first post in this thread proved the identity, actually.yungman said:
Evaluating Definite Integrals: A Comparison and Explanation
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SUMMARY
The discussion centers on evaluating the definite integrals of the form \(\int_{-\pi}^0 e^{jz\cos \theta}\cos(m\theta) d\theta\) and \(\int_0^{\pi} e^{jz\cos \theta}\cos(m\theta) d\theta\). Two substitution methods were applied: \(\theta = -\theta\) and \(\theta = \pi + \theta\), leading to different results that prompted confusion regarding their equality. The integrals are shown to be equal for specific values of \(m\), particularly \(m=0\), \(m=1\), and \(m=3\), where they yield the Bessel function \(J_m(z)\). The crux of the issue lies in the misunderstanding of the properties of definite integrals and the behavior of the integrands.
PREREQUISITES- Understanding of complex exponential functions and their properties.
- Familiarity with definite integrals and integration techniques.
- Knowledge of Bessel functions, specifically \(J_m(z)\).
- Ability to perform variable substitutions in integrals.
- Study the properties of Bessel functions, particularly \(J_m(z)\), and their applications in integral equations.
- Learn about the implications of variable substitution in definite integrals.
- Explore the relationship between complex exponentials and trigonometric functions in integrals.
- Investigate counterexamples in integral calculus to solidify understanding of equality in integrals.
Mathematicians, physics students, and anyone involved in advanced calculus or mathematical analysis, particularly those working with integrals and Bessel functions.
- #32
yungman
- 5,741
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Mandelbroth said:Your first post in this thread proved the identity, actually.
Yeh, hind sight show I was close! But I am not that good in math, still have problem twisting the substitution around every which way to see it.
It's one thing working through the class and textbooks, it's another thing to get into this Bessel function derivation that really twisting the substitution every which way! I am just glad I finally got my derivations for the Bessel functions needed for me to go back to my antenna theory. This and electromagnetics really take the calculus, ODE, PDE and numerical analysis through the ringer. Been stuck for two week in here struggling through this and I am only on chapter 5 of the antenna theory! One thing for sure, I will be a lot better off in math after this...and I don't know whether I am laughing or crying at this point!
- #33
andrien
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In another thread in which you asked about how to derive a different representation of bessel function,I told there that you should show it by yourself.The point is that only even m will contribute,all integrals involving odd m will vanish.For that you can use the expansion of cos[xcos(theta)].The thread is some below in this forum.
- #34
yungman
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andrien said:
What do you mean?
- #35
andrien
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I mean the thread is some below in this section of the forum.Also if you will use expansion of Sin(xCosθ),you will find that only odd m contribute and the extra minus sign outside Sin(xCosθ) will do the job.Just show it.yungman said:
- #36
yungman
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andrien said:
What do you mean by some below in this section?
- #37
andrien
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- #38
yungman
- 5,741
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andrien said:
Yes, I just visit and responded to the post. It's been a long road on this problem and I am glad I finally got my answer.
Thanks
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