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On Certain Definite Integrals And Infinite Series.pdf

Kummer__On_certain_definite_integrals_and_infinite_Series.pdf
Preview of On certain definite integrals and infinite series
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📊 Size: 149 KB
📄 Pages: 20 pages
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Summary

Ernst Eduard Kummer discusses definite integrals and infinite series, introducing functions φ(α, β, x), ψ(α, x), and χ(α, β, x) to represent certain series. He derives transformations of these series and applies them to solve the integral y = ∫[0,∞) u^(α-1) e^(-u) e^(-x/u) du, obtaining the solution y = Π(α-1)ψ(1-α, x) + Π(-α-1)x^α ψ(1+α, x). He then deduces another integral, expressed in terms of series of the form φ(α, β, x), and derives the formula ∫[0,∞) u^(α-1) e^(-ux) / (1+u)^β du = Π(α-β-1) / Π(α-1) x^β φ(β, β-α+1, x) + Π(β-α-1) / Π(β-1) x^α * φ(α, α-β+1, x).

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Ernst Eduard Kummer discusses definite integrals and infinite series, introducing functions φ(α, β, x), ψ(α, x), and χ(α, β, x) to represent certain series.

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  • File Format: PDF
  • File Size: 149 KB
  • Pages: 20
  • Language: EN
  • Total Downloads: 136
  • Last Updated: 7 days ago

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