Fdtxjr

Context: For a $2pi$-periodic bounded function $f:mathbbRtomathbbC$, we define the complex Fourier coefficients of $f$ by
$$
hatf_k:=frac12piint_0^2pif(x)e^-ikx,dx.
$$
We call Fourier series of $f$ the formal series
$$
sum_k=-infty^inftyhatf_ke^ikx.tag1
$$
Now, it is easily shown that
$$
overlinehatf_k=hatoverlinef_-k.
$$
Hence if $f$ is real-valued then
$$
overlinehatf_k=hatf_-k.tag2
$$
Reading some notes, it is said that if $f$ is a real-valued function, then we can write
$$
hatf_k=frac12piint_0^2pif(x)cos(kx),dx-ifrac12piint_0^2pif(x)sin(kx),dx.tag3
$$
Putting
$$
a_k:=frac1piint_0^2pif(x)cos(kx),dx,\
b_k:=frac1piint_0^2pif(x)sin(kx),dx,
$$
where $a_k,b_k$ are called the real Fourier coefficients of $f$, it is said that in view of $(2)$, we have the relations
beginalign
hatf_0&=frac12a_0,\
hatf_k&=frac12(a_k-ib_k),\
a_k&=hatf_k+hatf_-k,tag4\
b_k&=i(hatf_k-hatf_-k),\
a_-k&=a_k,\
b_-k&=-b_k.
endalign
Finally it is said that if, once again, $f$ is real-valued, then we can write $(1)$ as the trigonometric series
$$
frac12a_0+sum_k=1^inftya_kcos(kx)+b_ksin(kx).tag5
$$
Question: Is it really essential that $f$ be real-valued for $(3)$, $(4)$ and $(5)$ to hold? It seems to me that everything holds even if $f$ is complex-valued...

Yes they hold assuming $a_k,b_k$ are complex, but the point is that they are real valued and thus by using (2) we can show that
$$
hat f_k e^ikx + hat f_-k e^-ikx = hat f_k e^ikx + (hat f_k e^ikx)^* = 2 mathrmRe f_k e^ikx = 2 mathrmRe Big frac12 (a_k-ib_k)(cos(kx)+ isin(kx)) Big = a_kcos(kx) + b_k sin(kx)
$$

We can also show that same thing holds if $a_k,b_k$ are complex:

beginalign
hat f_k e^ikx + hat f_-k e^-ikx &= frac12(a_k - ib_k) e^ikx + frac12(a_-k - ib_-k) e^-ikx\&= frac12(a_k - ib_k) e^ikx + frac12(a_k + ib_k) e^-ikx \&= frac12 a_k (e^ikx+e^-ikx) - frac12i b_k( e^ikx - e^-ikx) \&= a_k cos(kx) + b_ksin(kx)
endalign