160308

sinc function の積の積分(1)

WolframAlpha で計算。

∫(0,∞)sinc(x)dx = π / 2

∫(0,∞)sinc(x)sinc(x / 2)dx = π / 2

∫(0,∞)sinc(x)sinc(x / 2)sinc(x / 3)dx = π / 2

∫(0,∞)sinc(x)sinc(x / 2)…sinc(x / 4)dx = 1727 π / 3456

∫(0,∞)sinc(x)sinc(x / 2)…sinc(x / 5)dx = 20652479 π / 41472000

∫(0,∞)sinc(x)sinc(x / 2)…sinc(x / 6)dx = 2059268143 π / 4147200000

∫(0,∞)sinc(x)sinc(x / 2)…sinc(x / 7)dx = 24860948333867803 π / 50185433088000000

それぞれ以下で確認できる。

http://www.wolframalpha.com/input/?i=integrate+%5Bsin(x)+%2F+(x%5E1),+%7Bx,+0,+%5Cinfty%7D%5D

http://www.wolframalpha.com/input/?i=integrate+%5Bsin(x)+*+sin(x%2F2)+*+2+%2F+(x%5E2),+%7Bx,+0,+%5Cinfty%7D%5D

http://www.wolframalpha.com/input/?i=integrate+%5Bsin(x)+*+sin(x%2F2)+*+sin(x%2F3)+*+2+*+3+%2F+(x%5E3),+%7Bx,+0,+%5Cinfty%7D%5D

http://www.wolframalpha.com/input/?i=integrate+%5Bsin(x)+*+sin(x%2F2)+*+sin(x%2F3)+*+sin(x%2F4)+*+2+*+3+*+4+%2F+(x%5E4),+%7Bx,+0,+%5Cinfty%7D%5D

http://www.wolframalpha.com/input/?i=integrate+%5Bsin(x)+*+sin(x%2F2)+*+sin(x%2F3)+*+sin(x%2F4)+*+sin(x%2F5)+*+2+*+3+*+4+*+5+%2F+(x%5E5),+%7Bx,+0,+%5Cinfty%7D%5D

http://www.wolframalpha.com/input/?i=integrate+%5Bsin(x)+*+sin(x%2F2)+*+sin(x%2F3)+*+sin(x%2F4)+*+sin(x%2F5)+*+sin(x%2F6)+*+2+*+3+*+4+*+5+*+6+%2F+(x%5E6),+%7Bx,+0,+%5Cinfty%7D%5D

http://www.wolframalpha.com/input/?i=integrate+%5Bsin(x)+*+sin(x%2F2)+*+sin(x%2F3)+*+sin(x%2F4)+*+sin(x%2F5)+*+sin(x%2F6)+*+sin(x%2F7)+*+2+*+3+*+4+*+5+*+6+*+7+%2F+(x%5E7),+%7Bx,+0,+%5Cinfty%7D%5D