■双子素数定数と・・・(その20)

【2】∫(0,∞)Πsin(kx)/(kx)dx=?

  ∫(0,∞)sinc(x)=π/2

  ∫(0,∞)sinc(x)sinc(2x)dx=π/4

  ∫(0,∞)sinc(x)sinc(2x)sinc(4x)dx=π/8

  ∫(0,∞)sinc(x)sinc(2x)sinc(4x)sinc(8x)dx=π/16

  ∫(0,∞)sinc(x)=π/2

  ∫(0,∞)sinc(x)sinc(3x)dx=π/6

  ∫(0,∞)sinc(x)sinc(3x)sinc(9x)dx=π/18

  ∫(0,∞)sinc(x)sinc(3x)sinc(9x)sinc(27x)dx=π/54

 置換積分により

  ∫(0,∞)Πsin(x*n^i)/(x*n^i)dx=1/n^k∫(0,∞)sin(x)/(x)dx

すなわち,本質的に同じ積分となる.

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