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T. Amdeberhan
T. Amdeberhan
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Let $a_1,\dots,a_n, b$ be positive real numbers.

Question.* Is this true? $$\int_{-\infty}^{\infty}\frac{\sin(bx+a_1x+\cdots+a_nx)}{x}\prod_{j=1}^n\frac{\sin(a_jx)}{a_jx}\,\,dx=1.$$$$\int_{-\infty}^{\infty}\frac{\sin(bx+a_1x+\cdots+a_nx)}{x}\prod_{j=1}^n\frac{\sin(a_jx)}{a_jx}\,\,dx=\pi.$$ My most immediate quest is "why is it independent of $b$, in particular?"

Let $a_1,\dots,a_n, b$ be positive real numbers.

Question.* Is this true? $$\int_{-\infty}^{\infty}\frac{\sin(bx+a_1x+\cdots+a_nx)}{x}\prod_{j=1}^n\frac{\sin(a_jx)}{a_jx}\,\,dx=1.$$ My most immediate quest is "why is it independent of $b$, in particular?"

Let $a_1,\dots,a_n, b$ be positive real numbers.

Question.* Is this true? $$\int_{-\infty}^{\infty}\frac{\sin(bx+a_1x+\cdots+a_nx)}{x}\prod_{j=1}^n\frac{\sin(a_jx)}{a_jx}\,\,dx=\pi.$$ My most immediate quest is "why is it independent of $b$, in particular?"

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T. Amdeberhan
T. Amdeberhan
  • 43.2k
  • 5
  • 57
  • 217

"sinc-ing" integral

Let $a_1,\dots,a_n, b$ be positive real numbers.

Question.* Is this true? $$\int_{-\infty}^{\infty}\frac{\sin(bx+a_1x+\cdots+a_nx)}{x}\prod_{j=1}^n\frac{\sin(a_jx)}{a_jx}\,\,dx=1.$$ My most immediate quest is "why is it independent of $b$, in particular?"