We have proved already (§ 215) that the sum of the series 1+(m1)z+(m2)z2+ is (1+z)m=exp{mlog(1+z)}, for all real values of m and all real values of z between 1 and 1. If an is the coefficient of zn then |an+1an|=|mnn+1|1, whether m is real or complex. Hence (Ex. LXXX. 3) the series is always convergent if the modulus of z is less than unity, and we shall now prove that its sum is still exp{mlog(1+z)}, the principal value of (1+z)m.

It follows from § 236 that if t is real then ddt(1+tz)m=mz(1+tz)m1, 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(m1)(mn+1)zn(1+tz)mn. This formula still holds if t=0, so that ϕn(0)n!=(mn)zn.

Now, in virtue of the remark made at the end of § 164, we have ϕ(1)=ϕ(0)+ϕ(0)+ϕ(0)2!++ϕ(n1)(0)(n1)!+Rn, where Rn=1(n1)!01(1t)n1ϕ(n)(t)dt. But if z=r(cosθ+isinθ) then |1+tz|=1+2trcosθ+t2r21tr, and therefore |Rn|<|m(m1)(mn+1)|(n1)!rn01(1t)n1(1tr)nmdt<|m(m1)(mn+1)|(n1)!(1θ)n1rn(1θr)nm, where 0<θ<1; so that (cf. § 163) |Rn|<K|m(m1)(mn+1)|(n1)!rn=ρn, say. But ρn+1ρn=|mn|nrr, and so (Ex. XXVII. 6) ρn0, and therefore Rn0, as n. Hence we arrive at the following theorem.

Theorem. The sum of the binomial series 1+(m1)z+(m2)z2+ is exp{mlog(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.

Example XCVIII
1. Suppose m real. Then since log(1+z)=12log(1+2rcosθ+r2)+iarctan(rsinθ1+rcosθ), we obtain 0(mn)zn=exp{12mlog(1+2rcosθ+r2)}Cis{marctan(rsinθ1+rcosθ)}=(1+2rcosθ+r2)12mCis{marctan(rsinθ1+rcosθ)}, all the inverse tangents lying between 12π and 12π. In particular, if we suppose θ=12π, z=ir, and equate the real and imaginary parts, we obtain 1(m2)r2+(m4)r4=(1+r2)12mcos(marctanr),(m1)r(m3)r3+(m5)r5=(1+r2)12msin(marctanr).

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 0r<1 then 11324r2+13572468r4=1+r2+12(1+r2),12r135246r3+13579246810r5=1+r212(1+r2).

[Take m=12 in the last two formulae of Ex. 1.]

4. Prove that if 14π<θ<14π then cosmθ=cosmθ{1(m2)tan2θ+(m4)tan4θ},sinmθ=cosmθ{(m1)tanθ(m3)tan3θ+}, for all real values of m. [These results follow at once from the equations cosmθ+isinmθ=(cosθ+isinθ)m=cosmθ(1+itanθ)m.]

5. We proved (Ex. LXXXI. 6), by direct multiplication of series, that f(m,z)=(mn)zn, where |z|<1, satisfies the functional equation f(m,z)f(m,z)=f(m+m,z). Deduce, by an argument similar to that of § 216, and without assuming the general result above, that if m is real and rational then f(m,z)=exp{mlog(1+z)}.

6. If z and μ are real, and 1<z<1, then (iμn)zn=cos{μlog(1+z)}+isin{μlog(1+z)}.


236. The exponential limit Main Page MISCELLANEOUS EXAMPLES ON CHAPTER X