|β′μ| < g / ρμ 2m2
provided (μ + m2 + 1)μ < (3⁄2)m2 + 1; we take m2 = n2n − 2, supposing μ < 2n2n−4. So long as λ2 ⋜ m2 ⋜ n2n−2 and μ < 2n2n−4 we have μ + λ2 < 2n2n−2, and we can use the previous inequality to substitute here for φ(μ + λ2) (a1). When this is done we find
| φ(μ) (a2) = Σ m2λ2=0Σ m1λ1=0 | φ(μ + λ1 + λ2) (0) | ( | x | ) λ1 + λ2 + βμ , |
| λ1! λ2! | n |
where |βμ| < 2g/ρμ 2m2, the numbers m1, m2 being respectively n2n and n2n−2.
Applying then the original inequality to φ(μ) (a3) = φ(μ) (a2 + x/n), and then using the series just obtained, we find a series for φ(μ) (a3). This process being continued, we finally obtain
| φ(x) = Σ m1λ1=0Σ m2λ2=0 ... Σ mnλn=0 | φh (0) | ( | x | ) h + ε , |
| K | n |
where h = λ1 + λ2 + ... + λn, K = λ1! λ2! ... λn!, m1 = n2n, m2 = n2n−2, ..., mn= n², |ε| < 2g/2mn.
By this formula φ(x) is represented, with any required degree of accuracy, by a polynomial, within the region in question; and thence can be expressed as before by a series of polynomials converging uniformly (and absolutely) within this region.
§ 13. Application of Cauchy’s Theorem to the Determination of Definite Integrals.—Some reference must be made to a method whereby real definite integrals may frequently be evaluated by use of the theorem of the vanishing of the integral of a function of a complex variable round a contour within which the function is single valued and non singular.
We are to evaluate an integral ∫ ba ƒ(x)dx; we form a closed contour of which the portion of the real axis from x = a to x = b forms a part, and consider the integral ∫ƒ(z)dz round this contour, supposing that the value of this integral can be determined along the curve forming the completion of the contour. The contour being supposed such that, within it, ƒ(z) is a single valued and finite function of the complex variable z save at a finite number of isolated interior points, the contour integral is equal to the sum of the values of ∫ƒ(z)dz taken round these points. Two instances will suffice to explain the method. (1) The integral ∫ ∞0 [(tan x)/x] dx is convergent if it be understood to mean the limit when ε, ζ, σ, ... all vanish of the sum of the integrals