Bessel Function of the 2nd Kind | 2nd solution of Bessel's Equation

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In this video, I briefly describe how to obtain the Bessel function of the second kind, which is also the 2nd solution to Bessel's equation when the order of the ODE is an integer. People requested this video in the past so I hope this video is sufficient!

Deriving Y_n is fairly simple: take a particular linear combination of J_p and J_-p, and then let p approach n. I don't go through the entire calculation, because it's tedious and doesn't really teach much, but I refer you to this document (by Jim Lambers at Berkeley) if you're interested in the details:

Questions/requests? Let me know in the comments!

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You did in 5 minutes what my professor couldn't in half of a week: Explaining it clearly. Thanks!

JoshuaSalazarMejia
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my problem is where this special linear combination came from
my book doesn't explain it either

bebarshossny
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Appreciate your effort. Very nicely explained

gautamhathiwala
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Thanks a lot for your great videos, they are extremely helpfull. I have 1 question about the power of x in Jn (minute 5:00), here its 2k+n, but in the Jp's ive seen you define so far, this power stops at 2k. Why is this, or did i find a rare error? ps. the link to Jim Lambers document is not working.

VonnBellum
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Hi! Loving these videos but small question: at 3:50, why is J(-n)=(-1)^nJ(n)? I couldn't figure out where this property comes from. Thank you very much in advance.

qirunwei
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exceptional approach talent.. dear sir, u r 1 of a kind, truly remarkable.. all the best for ur future shows..

rummanmuhammad
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Oh, it wasn't LA hospital rules... my bad

tradetor
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Thank you so much I'm glad I found your channel !

yeonsulee
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Is for the first summation (n-1) not infinity?

virgilpanfil
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How can anybody use something so long? It is disappointing to come all this way with you and not be able to finish.

the_eternal_student
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Hi could you prove the hypothesis that "Potatoes are the best thing in this world"

MBAzeBest