It follows from 236 that if t is real then d dt 1 tz

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It follows from § 236 that if t is real then d dt (1 + tz ) m = mz (1 + tz ) m - 1 , z and m having any real or complex values and each side having its principal value. Hence, if φ ( t ) = (1 + tz ) m , we have φ ( n ) ( t ) = m ( m - 1) . . . ( m - n + 1) z n (1 + tz ) m - n . This formula still holds if t = 0, so that φ n (0) n ! = m n z n . Now, in virtue of the remark made at the end of § 164 , we have φ (1) = φ (0) + φ 0 (0) + φ 00 (0) 2! + · · · + φ ( n - 1) (0) ( n - 1)! + R n , where R n = 1 ( n - 1)! Z 1 0 (1 - t ) n - 1 φ ( n ) ( t ) dt.
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[X : 237] THE GENERAL THEORY OF THE LOGARITHMIC, 532 But if z = r (cos θ + i sin θ ) then | 1 + tz | = 1 + 2 tr cos θ + t 2 r 2 = 1 - tr, and therefore | R n | < | m ( m - 1) . . . ( m - n + 1) | ( n - 1)! r n Z 1 0 (1 - t ) n - 1 (1 - tr ) n - m dt < | m ( m - 1) . . . ( m - n + 1) | ( n - 1)! (1 - θ ) n - 1 r n (1 - θr ) n - m , where 0 < θ < 1; so that (cf. § 163 ) | R n | < K | m ( m - 1) . . . ( m - n + 1) | ( n - 1)! r n = ρ n , say. But ρ n +1 ρ n = | m - n | n r r, and so ( Ex. xxvii . 6) ρ n 0, and therefore R n 0, as n → ∞ . Hence we arrive at the following theorem. Theorem. The sum of the binomial series 1 + m 1 z + m 2 z 2 + . . . is exp { m log(1 + z ) } , where the logarithm has its principal value, for all values of m , real or complex, and all values of z such that | z | < 1 . A more complete discussion of the binomial series, taking account of the more difficult case in which | z | = 1, will be found on pp. 225 et seq. of Bromwich’s Infinite Series . Examples XCVIII. 1. Suppose m real. Then since log(1 + z ) = 1 2 log(1 + 2 r cos θ + r 2 ) + i arc tan r sin θ 1 + r cos θ ,
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[X : 237] EXPONENTIAL, AND CIRCULAR FUNCTIONS 533 we obtain X 0 m n z n = exp { 1 2 m log(1 + 2 r cos θ + r 2 ) } Cis m arc tan r sin θ 1 + r cos θ = (1 + 2 r cos θ + r 2 ) 1 2 m Cis m arc tan r sin θ 1 + r cos θ , all the inverse tangents lying between - 1 2 π and 1 2 π . In particular, if we suppose θ = 1 2 π , z = ir , and equate the real and imaginary parts, we obtain 1 - m 2 r 2 + m 4 r 4 - . . . = (1 + r 2 ) 1 2 m cos( m arc tan r ) , m 1 r - m 3 r 3 + m 5 r 5 - . . . = (1 + r 2 ) 1 2 m sin( m arc tan r ) . 2. Verify the formulae of Ex. 1 when m = 1, 2, 3. [Of course when m is a positive integer the series is finite.] 3. Prove that if 0 5 r < 1 then 1 - 1 · 3 2 · 4 r 2 + 1 · 3 · 5 · 7 2 · 4 · 6 · 8 r 4 - . . . = s 1 + r 2 + 1 2(1 + r 2 ) , 1 2 r - 1 · 3 · 5 2 · 4 · 6 r 3 + 1 · 3 · 5 · 7 · 9 2 · 4 · 6 · 8 · 10 r 5 - . . . = s 1 + r 2 - 1 2(1 + r 2 ) . [Take m = - 1 2 in the last two formulae of Ex. 1.] 4. Prove that if - 1 4 π < θ < 1 4 π then cos = cos m θ 1 - m 2 tan 2 θ + m 4 tan 4 θ - . . . , sin = cos m θ m 1 tan θ - m 3 tan 3 θ + . . . , for all real values of m . [These results follow at once from the equations cos + i sin = (cos θ + i sin θ ) m = cos m θ (1 + i tan θ ) m . ]
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[X : 237] THE GENERAL THEORY OF THE LOGARITHMIC, 534 5. We proved ( Ex. lxxxi . 6), by direct multiplication of series, that f ( m, z ) = m n z n , where | z | < 1, satisfies the functional equation f ( m, z ) f ( m 0 , z ) = f ( m + m 0 , z ) . Deduce, by an argument similar to that of § 216 , and without assuming the general result of p. 532 , that if m is real and rational then f ( m, z ) = exp { m log(1 + z ) } . 6. If z and μ are real, and - 1 < z < 1, then X n z n = cos { μ log(1 + z ) } + i sin { μ log(1 + z ) } .
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