To proove the inequality:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\left | \int_a^b f(t) dt \right | \le \int_a^b | f(t) | dt[/tex]

for complex valued [tex]f[/tex], use the following:

[tex]\textrm{Re}\left [ e^{-\iota\theta}\int_a^b f(t) dt\right ] = \int_a^b \textrm{Re}[e^{-\iota\theta}f(t)] dt \le \int_a^b | f(t) | dt[/tex]

and then if we set:

[tex]\theta = \textrm{arg}\int_a^b f(t) dt[/tex]

the expression on the left reduces to the absolute value [tex]\square[/tex]

So the last part, where he says that it reduces to the modulus by using that substitution of theta, I don't get it... how does that reduce to the modulus?

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# Fundamental complex inequality

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