# Fourier Series of Cosh

I'm trying to calculate the Fourier Series of Cosh[gS[[2]] z]. In the Range of (-b, b) In order to do that, I'm using the function CoshFou I created. Also I'm using the following paramters:

b = 500
gS = {0.157121, 0.31418, 0.471253, 0.628329, 0.785406, 0.942485, 1.09956, 1.25664, 1.41372}
nMax = 8


The function CoshFou:

CoshFou[z_, s_] := Block[{a0, aN, Fun},
a0 = 2/(2 b) NIntegrate[Cosh[gS[[s]] zz], {zz, -b, b}];
aN = Table[ 2/(2 b) NIntegrate[Cosh[gS[[s]] zz] Cos[(2 Pi nIt zz)/(2 b)], {zz, -b, b}], {nIt, nMax}];
Fun = a0/2 + Sum[aN[[nIt]] Cos[(2 Pi nIt z) / (2 b)], {nIt,  nMax - 1}]
];


In this function, a0 and aN are the Fourier coefficients and Fun is the actual Series. When I'm plotting the Fourier series for gS[[2]] and check against the Cosh function provided by Mathematica I see a big difference and I can't explain why this big difference exists.

I my mind I apply the Fourier Transformation correctly and I know that it is more accurate when I use more Series elemets but I tried more and the results don't get better.

• I think this is a result of two things: First, your sum should include the term for nIt=1, and second, you need a lot more terms for a good approximation of a function with such a steep rise towards the edge. For example, nMax=2500;b=300 gives good results, while changing b to 350 or 400 starts to show ringing artifacts – Lukas Lang Dec 11 '20 at 8:35
• @LukasLang Thanks for the hint. The 2 was from a previous iteration of the function. I removed it and now it looks better. For the series elements I still don't get why my maximum amplitude is at 10^66 and the Max and the Series is at 10^56 and why it should reduce with more elements. – CR36 Dec 11 '20 at 14:05

With the given function I can plot:

CoshFou[z_, s_] :=
Block[{a0, aN, Fun},
a0 = 2/(2 b) NIntegrate[Cosh[gS[[s]] zz], {zz, -b, b}];
aN = Table[
2/(2 b) NIntegrate[
Cosh[gS[[s]] zz] Cos[(2 Pi nIt zz)/(2 b)], {zz, -b, b}], {nIt,
nMax}];
Fun = a0/2 +
Sum[aN[[nIt]] Cos[(2 Pi nIt z)/(2 b)], {nIt, nMax - 1}]];

Plot[Sum[CoshFou[z, s], {s, 1, Length@gS}], {z, -1, 1}]


This shows already the rapid divergence and figures up to $$10^303$$.

For a proper definition of the Fourier cosine series look up FourierCosSeries.

The function is a sum of cosine hyperbolicus at different frequencies. This can not be meant as frequencies of the coefficients of the Fourier cosine series. Mathematica allows to make use of the nMax parameter in the built-in function. There is no such periodic meaning like a cosine hyperbolicus transform.

FourierCosSeries[Sum[Cosh[gS[[s]] t], {s, 1, Length@gS}], t, 8]


(* 31.724 - (32.4347 - 3.2029810^-16 I) Cos[ t] + (14.5598 - 1.4135810^-16 I) Cos[ 2 t] - (7.65296 + 3.9007910^-16 I) Cos[ 3 t] + (4.60424 + 1.2913110^-16 I) Cos[ 4 t] - (3.04553 - 9.6337610^-16 I) Cos[ 5 t] + (2.15436 - 1.7638910^-16 I) Cos[ 6 t] - (1.60082 - 4.3831510^-16 I) Cos[ 7 t] + (1.23477 + 1.9865610^-16 I) Cos[8 t] *)

ReImPlot[%, {t, -\[Pi], \[Pi]}]


For the calculation of such coefficients there is not the real big interval important but the periodizity of the cosine function. So the coefficient are not calculated on the large interval but on the periodizity interval of the cosine. That keeps the the coefficients small and the integration short. To be valid on the large interval extrapolation can be hoped. In most cases this is for divergent function not the case. The Fourier transforms make the functions periodic.

If the series is dropped there is the integral Fourier transform: FourierTransform. FourierTransform[Sum[Cosh[gS[[s]] t], {s, 1, Length@gS}], t, \[Omega]] (* fails *)

that makes use of the infinite real axis and thereby for approximations big intervals.

There is main value definition: $$\sqrt{\frac{\vert b\vert}{(2\pi)^{1-a}}}\int_{-\infty}^{\infty}f(t)e^{i b \omega t}dt$$

Integrate[
Sum[Cosh[gS[[s]] t], {s, 1, Length@gS}] Exp[i b \[Omega] t], {t, -b,
b}]

(* (-(9.669354397782823*^303/(-0.00282744 + 1.*i*\[Omega])) - 7.514651931406424*^269/
(-0.00251328 + 1.*i*\[Omega]) - 5.840099693021041*^235/(-0.00219912 + 1.*i*\[Omega]) -
4.550062773378836*^201/(-0.0018849700000000001 + 1.*i*\[Omega]) -
3.5379030434103284*^167/(-0.001570812 + 1.*i*\[Omega]) +
(1.9733621830656194*^179 + 3.140496185412215*^182*i*\[Omega] -
2.345487709898561*^186*i^2*\[Omega]^2 - 3.732713269301929*^189*i^3*\[Omega]^3 +
3.655743265116067*^192*i^4*\[Omega]^4 + 5.817912128582447*^195*i^5*\[Omega]^5 -
1.424544082331069*^198*i^6*\[Omega]^6 - 2.2670826951605273*^201*i^7*\[Omega]^7)/
(5.46950432817499*^-26 - 3.5264830524668625*^-38*i*\[Omega] -
7.886171420873058*^-19*i^2*\[Omega]^2 - 2.9582283945787943*^-31*i^3*\[Omega]^3 +
2.659733639289811*^-12*i^4*\[Omega]^4 + 6.203854594147708*^-25*i^5*\[Omega]^5 -
2.9610912131639997*^-6*i^6*\[Omega]^6 + 2.168404344971009*^-19*i^7*\[Omega]^7 + 1.*i^8*\[Omega]^8) +
E^(500000.*i*\[Omega])*(3.5379030434103284*^167/(0.001570812 + 1.*i*\[Omega]) +
4.550062773378836*^201/(0.0018849700000000001 + 1.*i*\[Omega]) +
5.840099693021041*^235/(0.00219912 + 1.*i*\[Omega]) + 7.514651931406424*^269/
(0.00251328 + 1.*i*\[Omega]) + 9.669354397782823*^303/(0.00282744 + 1.*i*\[Omega])))/
E^(250000.*i*\[Omega]) *)


This has to be multiplicated with the leading factor still.

if $$\infty$$ if shortened to $$b$$ this makes sense. But for that nMax is meaningless. From this approximated integral the nMax gets the meaning back. Some common choices for {a,b} are {0,1} (default; modern physics), {1,-1} (pure mathematics; systems engineering), {-1,1} (classical physics), and {0,-2Pi} (signal processing).

• Thanks for your answer Steffen. Maybe I wasn't exact on that point, but I don't want theSum of the function CoshFou. I want to express the function Cosh[z, gs[[s]]]  as a Fourier-Series. So I want to have 8 Fourier-Series in the end. I wanted to use FourierCosSeries at first but I don't understand the documentation of this function I will dive into it right away. – CR36 Dec 14 '20 at 7:05
• Cosh[z, gs[[s]]]  does not work. (Cosh)[reference.wolfram.com/language/ref/Cosh.html] is a univariate. Just one value in the square brackets. – Steffen Jaeschke Dec 18 '20 at 15:04
• FourierCosSeries[Cosh[gS[[2]] z], z, nMax] is the built-in for You. Whether You use nMax or a smaller value in not important. Cosh has no finite FourierCosSeries. Mind there is always a real and a complex finite FourierCosSeries. Hyperbolics and Trigonometrics do not fit well together. Here is a nice solution: (fourier seriesfor cosh)[math.stackexchange.com/questions/1970922/… – Steffen Jaeschke Dec 18 '20 at 15:23