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Chapter 9
numerator and the second expansion to the square root in the denominator. Thus, we are multiplying together three infinite series. This results in calculations too lengthy to reproduce here, but Makota did summarize them on the tablet. First, he writes the three infinite series in terms of the following recursion formulas:
(1) With s ≡ 1 − b 2/a 2, a and b as above, let 1 c 1 ≡ (1 − s ), 2 1 c 2 ≡ (3 + s )c 1 4 1 ⎛ 1⎞ c 3 ≡ (5 + 3s )c 2 − ⎜ ⎟ sc 1, ⎝ 3⎠ 6 c4 ≡
1 ⎛ 2⎞ (7 + 5 s )c 3 − ⎜ ⎟ sc 2 , ⎝ 4⎠ 8
1 ⎛ 3⎞ (9 + 7s )c 4 − ⎜ ⎟ sc 3 , ⎝5 ⎠ 10 ...≡.... c5 ≡
(2) With w ≡ d 2/a 2 and c ≡ D 2 − (D − 2d )2 , let c3 , D B 2 ≡ wB1, B1 ≡
B 3 ≡ wB 2 , B4 ≡ wB 3 , ...≡....
(3) With t ≡ d ·D/a 2 and k ≡ D/d, let ⎛ 1⎞ A0 ≡ 2S 1, D1 ≡ ⎜ ⎟ ⎝ 8⎠
5 ⎛ ⎞ t ⎜ 3 A0 − kB1 − B1 ⎟ , ⎝ ⎠ 6
9 ⎛ 1⎞ ⎛ ⎞ A1 ≡ c 1D1, D 2 ≡ ⎜ ⎟ t ⎜ 7 D1 − kB2 − B 2 ⎟ , ⎝ 12 ⎠ ⎝ ⎠ 10 13 ⎛ 1⎞ ⎛ ⎞ A2 ≡ c 2 D 2 , D 3 ≡ ⎜ ⎟ t ⎜ 11D 2 − kB3 − B 3 ⎟ , ⎝ 16 ⎠ ⎝ ⎠ 14 ⎛ 1⎞ A3 ≡ c 3 D 3 , D4 ≡ ⎜ ⎟ ⎝ 20 ⎠
17 ⎛ ⎞ t ⎜ 15 D 3 − kB4 − B4 ⎟ , ⎝ ⎠ 18
21 ⎛ 1⎞ ⎛ ⎞ A4 ≡ c 4 D4 , D5 ≡ ⎜ ⎟ t ⎜ 19D4 − kB5 − B5 ⎟ , ⎝ 24 ⎠ ⎝ ⎠ 22 ...≡....