Abstract
In Sect. I.7.2 of the first book we investigated in detail how an arbitrary function f(x) can be expanded into a functional series in terms of functions a n (x) with n = 0, 1, 2, …, i.e.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
In the following, references to the first volume of this course (L. Kantorovich, Mathematics for natural scientists: fundamentals and basics, Springer, 2015) will be made by appending the Roman number I in front of the reference, e.g. Sect. I.1.8 or Eq. (I.5.18) refer to Sect. 1.8 and Eq. (5.18) of the first volume, respectively.
- 2.
Or, which is the same, which is periodic with the period of T = 2l; the “main” or “irreducible” part of the function, which is periodically repeated, can start anywhere, one choice is between − l and l, the other between 0 and 2l, etc.
- 3.
Since the Fourier series for both y(t) and f(t) converge, the series for \(\ddot{y}(t)\) must converge as well, i.e. the second derivative of the Fourier series of y(t) must be well defined.
- 4.
The Ewald method corresponds to a particular regularisation of the conditionally converging series. However, it can be shown (see, e.g., L. Kantorovich and I. Tupitsin—J. Phys. Cond. Matter 11, 6159 (1999)) that this calculation results in the correct expression for the electrostatic potential in the central part of a large finite sample if the dipole and quadruple moments of the unit cell are equal to zero. Otherwise, an additional macroscopic contribution to the potential is present.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Kantorovich, L. (2016). Fourier Series. In: Mathematics for Natural Scientists II. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-27861-2_3
Download citation
DOI: https://doi.org/10.1007/978-3-319-27861-2_3
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-27859-9
Online ISBN: 978-3-319-27861-2
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)