Can someone please show/explain to me the steps between the 2 circled formulas on the attached page #582 from Riley, Hobson, Bence - Mathematical Methods for Physics and Engineering 3rd edition.
We can write the recurrence relation as K_l = \frac{2l}{2l+1} K_{l-1}.
We get the formula on the left-hand side when we substitute K_{l-1} with K_{l-1} = \frac{2l-1}{2(l-1)+1} K_{l-1-1}.
We can repeat this process until we get to l = 1 and K_0 (because of the assumption just below the grey box).
The part most on the right of the circle below is a compact way to write this product.
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#3
vgarg
11
0
Thank you!
Could you please explain where does the 2nd ## 2^l l! ## term in ## 2^l l! \frac{2^l l!}{(2l+1)!} 2 ## in the lower circle come from? It has two ## 2^l l! ## terms in the numerator.
#4
dirichlet
3
2
I don't know exactly how they arrive at that expression, but it could be due to some conversion that is related to the double factorial.
See: https://en.wikipedia.org/wiki/Double_factorial
#5
vgarg
11
0
Thank you!
Can someone else please try to explain this to me?