Let be $f:\mathbb{R}\rightarrow \mathbb{C}$ Why $|\int_{-\infty}^{\infty}f(x)dx|\leq\int_{-\infty}^{\infty}|f(x)|dx$?
pdta:$|\cdot|$ is module of complex numbers
Let be $f:\mathbb{R}\rightarrow \mathbb{C}$ Why $|\int_{-\infty}^{\infty}f(x)dx|\leq\int_{-\infty}^{\infty}|f(x)|dx$?
pdta:$|\cdot|$ is module of complex numbers
Note that $\left| \int_a^b f(x)dx \right| = e^{-i\theta}\int_a^b f(x)dx = \int_a^b e^{-i\theta}f(x)dx$ where $\theta = \operatorname{Arg}\int_a^b f(x)dx$. But $\left| \int_a^b f(x)dx \right|$ is real, so $\left| \int_a^b f(x)dx \right| = \operatorname{Re}\int_a^b e^{-i\theta}f(x)dx = \int_a^b \operatorname{Re}(e^{-i\theta}f(x))dx \leq \int_a^b \left| e^{-i\theta}f(x) \right|dx = \int_a^b \left| f(x) \right|dx.$