How to approach to the Bessel function of second kind definition?

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We know that we can find the Bessel function of first kind by using power series approach. My question is How to find the Bessel function of second kind?

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You can find it also using a power series approach.

Section 5.5 of Advanced Engineering Mathematics 10th Edition by Erwin Kreyszig walks you through the derivation. You will need to read sections 5.3 and 5.4 to understand all the steps though.

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