I'm beginning my way through Coddington's Intro to ODE's and I'm a little thrown off in the preliminary section in a proof regarding complexed valued functions. ( I should note that I've taken a course in ODES, but my background in Complex anything-besides-the-basics is subpar. )
Particularly the book shows that:
$ \left| \int_a^b f(x) \, dx \right| \leq \int_a^b \left| f(x) \right| \, dx $
The proof is pretty short, so I guess I'll just map it out until the me-thrown-off point.
First, let
$ F \, = \, \int_a^b f(x) \, dx \quad $
and
$ u = \cos(\theta) + i\sin(\theta). $
Then, let $ F \, = \, \left| F \,\right| u $ where $ F \neq 0 $.
Since $u\overline{u} = 1$,
$ \left| F \right| = \,\overline{u} F \, = \;\overline{u} \int_a^b f(x) \, dx = \;...$
and the step that loses me:
$ ...\; = \, Re\left[ \; \overline{u} \int_a^b f(x) \, dx \; \right] \, = \; ... $
Maybe my unfamiliarity with complex-valued functions is making me miss something obvious, but I'm stumped. For instance, I've tried expanding $ \overline{u} \int_a^b f(x) \, dx $ to:
$ ( \cos(\theta) - i \sin(\theta) ) \int_a^b f(x) \, dx $ $ = \; \cos(\theta) \int_a^b f(x) \, dx - i sin(\theta) \int_a^b f(x) \, dx $ $ = \; \cos(\theta) \left( \int_a^b \left ( Re \, f \, \right) (x) \, dx + i \int_a^b \left ( Im \, f \, \right) (x) \, dx \right)- i sin(\theta) \left( \int_a^b \left ( Re \, f \, \right) (x) \, dx + i \int_a^b \left ( Im \, f \, \right) (x) \, dx \right) $ $ = \; \cos(\theta) \int_a^b \left ( Re \, f \, \right) (x) \, dx + i \cos(\theta) \int_a^b \left ( Im \, f \, \right) (x) \, dx - i sin(\theta) \int_a^b \left ( Re \, f \, \right) (x) \, dx + sin(\theta) \int_a^b \left ( Im \, f \, \right) (x) \, dx $ $ = \; \overline{u} \int_a^b \left( Re \, f \right) (x) \, dx + \left( i \cos(\theta) + \sin(\theta)\right) \int_a^b \left ( Im \, f \, \right) (x) \, dx $
I felt as if I was on the right track but I hit a wall at this point, and it began to feel like I was convoluting something simple. Anyway the proof finishes with:
$ ... \; = \; \int_a^b Re\left[ \overline{u} f(x) \right] \, dx \; \leq \int_a^b \left| f(x) \right| \, dx $
Any help is appreciated :)