Section 5.1 Direchlet’s Conditions
Direchlet’s Conditions are a set of conditions that need to be satisfied by a function in order for its Fourier series representation to converge to the function itself. In other words, to apply Fourier series representation of any function \(f(x)\text{,}\) it is necessary that \(f(x)\) must satisfy the following conditions.
- Periodicity: The function f(x) must be periodic, meaning it repeats itself over a specific interval. The period can be any finite value or infinite. The Fourier series is defined specifically for periodic functions. That is, \(f(x)\) is a singled - valued within the period T.
- Finite Number of Discontinuities: \(f(x)\) has a finite number of discontinuities in the period T. Discontinuities can include jump discontinuities or removable discontinuities (such as isolated points). The presence of an infinite number of discontinuities or essential discontinuities can prevent the Fourier series from converging to the function.
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Finite Total Variation: The function f(x) must have a finite total variation within a given period. The total variation of a function measures the overall "wiggliness" or oscillation of the function. If the total variation is infinite, the Fourier series may not converge. That is, \(f(x)\) has a finite number of maxima and minima in the period T.and \(f(x)\) is absolutely integrable. i.e.,\begin{equation*} \int\limits_{-T/2}^{T/2} |f(x)|\, dx \lt \infty, \end{equation*}or, a series must be convergent.
If a function satisfies these three conditions, then its Fourier series representation converges to the function pointwise, except at the points of discontinuity where the series converges to the average of the left and right limits. It is important to note that Dirichlet’s conditions are necessary but not sufficient for the convergence of the Fourier series. There exist functions that satisfy these conditions but whose Fourier series does not converge at specific points or over the entire interval. Additional conditions, such as the smoothness of the function, are required to ensure convergence in those cases.
Some Useful Formulae:.
- \begin{equation*} \int\limits_{0}^{2\pi} \sin nx \, dx = 0. \end{equation*}
- \begin{equation*} \int\limits_{0}^{2\pi} \cos nx \, dx = 0. \end{equation*}
- \begin{equation*} \int\limits_{0}^{2\pi} \sin^{2} nx\, dx = \pi. \end{equation*}
- \begin{equation*} \int\limits_{0}^{2\pi} \cos nx \,dx = \pi. \end{equation*}
- \begin{equation*} \int\limits_{0}^{2\pi} \sin nx .\sin mx\, dx = 0. \end{equation*}
- \begin{equation*} \int\limits_{0}^{2\pi} \cos nx .\cos mx\, dx = 0. \end{equation*}
- \begin{equation*} \int\limits_{0}^{2\pi} \sin nx .\cos mx\, dx = 0. \end{equation*}
- \begin{equation*} \int\limits_{0}^{2\pi} \sin nx .\cos nx\, dx = 0 \end{equation*}
- \begin{equation*} \sin n\pi = 0. \end{equation*}
- \begin{equation*} \cos n\pi = (-1)^{n}, \quad \text{where n is any integer}. \end{equation*}
- \begin{equation*} u v\, dx = u v_{1}-u' v_{2}+u''v_{3}-\cdots \end{equation*}where \(u' = \frac{\,du}{\,dx}, u'' = \frac{\,d^{2}u}{\,dx^{2}}, \cdots;\) \(v_{1}=\int v\, dx, v_{2}= \int v_{1}\,dx, \cdots\text{.}\)
- \begin{equation*} \int e^{ax} \cos bx\, dx = \frac{e^{ax}}{a^{2}+b^{2}}(a\cos bx+b\sin bx) \end{equation*}
- \begin{equation*} \int e^{ax} \sin bx\, dx = \frac{e^{ax}}{a^{2}+b^{2}}(a\sin bx-b\cos bx) \end{equation*}
- \begin{equation*} \int\limits_{-\pi}^{\pi} \sin nx .\sin mx\, dx \end{equation*}\begin{equation*} = \begin{cases} =0, \quad m=0 =\pi \delta_{mn}, \quad m\neq 0 \end{cases} \end{equation*}
- \begin{equation*} \int\limits_{-\pi}^{\pi} \sin nx .\cos mx\, dx = 0. \end{equation*}
- \begin{equation*} \int\limits_{-\pi}^{\pi} \cos nx .\cos mx\, dx = \begin{cases} =2\pi, \quad m==n=0\\ =\pi \delta_{mn}, \quad m\neq 0 \end{cases} \end{equation*}
- \begin{equation*} \sin A\sin B = \frac{1}{2}\left[\cos (A-B) - \cos(A+B)\right]; \end{equation*}
- \begin{equation*} \cos A\cos B = \frac{1}{2}\left[\cos (A-B) + \cos(A+B)\right]; \end{equation*}
- \begin{equation*} \sin A\cos B = \frac{1}{2}\left[\sin (A-B) + \sin (A+B)\right]. \end{equation*}
