Is the hyperbolic sine Fourier series a convergent series?

 Is the hyperbolic sine Fourier series a convergent series?

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The convergence of a Fourier series depends on the properties of the function being represented. The hyperbolic sine function, $\sinh (𝑥)$, is an odd and unbounded function, defined as:

$$sinh⁡(𝑥)=𝑒𝑥−𝑒−𝑥2

To determine if the Fourier series of $\sinh (𝑥)$ converges, we need to analyze how the Fourier series is defined for this particular function.

Fourier Series on Finite Interval

If we consider the Fourier series representation of $\sinh (𝑥)$ on a finite interval $[-𝐿,𝐿]$, we express the function in terms of sines and cosines (or equivalently in complex exponential form):

$\sinh (𝑥)\approx {\sum }_{𝑛=-\mathrm{\infty }}^{\mathrm{\infty }}{𝑐}_{𝑛}{𝑒}^{𝑖\frac{𝑛𝜋𝑥}{𝐿}}$

where the coefficients ${𝑐}_{𝑛}$ are given by:

${𝑐}_{𝑛}=\frac{1}{2𝐿}{\int }_{-𝐿}^{𝐿}\sinh (𝑥){𝑒}^{-𝑖\frac{𝑛𝜋𝑥}{𝐿}}𝑑𝑥$

Convergence on Finite Interval

On a finite interval $[-𝐿,𝐿]$, the Fourier series for $\sinh (𝑥)$ will converge, typically in the mean square sense (L² convergence). The convergence properties can be examined more closely using results from Fourier analysis:

1. Piecewise Continuity: $\sinh (𝑥)$ is continuous and differentiable everywhere.

2. Square Integrability: On any finite interval $[-𝐿,𝐿]$, $\sinh (𝑥)$ is square integrable because:

   ${\int }_{-𝐿}^{𝐿}{\mid \sinh (𝑥)\mid }^2𝑑𝑥<\mathrm{\infty }$

Given these properties, the Fourier series of $\sinh (𝑥)$ on a finite interval will converge to $\sinh (𝑥)$ at almost every point within that interval, and uniformly on any closed subinterval within $[-𝐿,𝐿]$.

Fourier Series on Infinite Interval

If we were to consider the Fourier series representation of $\sinh (𝑥)$ on the whole real line (which is less common), we'd typically use the Fourier transform rather than a Fourier series because $\sinh (𝑥)$ is unbounded as $𝑥\to \pm \mathrm{\infty }$.

Conclusion

For the practical purposes of most applications involving Fourier series (such as signal processing or solving differential equations on finite intervals), the Fourier series representation of $\sinh (𝑥)$ on a finite interval $[-𝐿,𝐿]$ is convergent.

Therefore, the hyperbolic sine Fourier series is convergent on any finite interval.