tailieunhanh - Handbook of mathematics for engineers and scienteists part 57

The center O-y of the incircle (the incemer) is the point where the angle bisectors meet (Fig. ). The straight lines connecting the vertices of a triangle with the points at which the incircle is tangent to the respective opposite sides intersect in a single point G called the Gergonne point (Fig. ). | 360 Series . Fourier expansion of odd and even functions. 1 . Let f x be an even function . f x f -x . Then the Fourier expansion of f x on the interval -l l has the form of the cosine Fourier series œ f x an cos n 1 where the Fourier coefficients have the form an 2 Î f x cos nnx dx bn 0 . l Jo l 2 . Let f x be an odd function . f x -f -x . Then the Fourier expansion of f x on the interval -l l has the form of the sine Fourier series œ f x bn sin x n 1 where the Fourier coefficients have the form 2 ft nnx bn j J f x sin dx an 0 . Example. Let us find the Fourier expansion of the function f x x on the interval -n n . Taking l n and f x x in the formula for the Fourier coefficients and integrating by parts we obtain bn x sin nx dx - x cos nx l cos nx dx - cos nn -1 n 1 . n J0 n y n Io n J0 J n n Therefore the Fourier expansion of f x x has the form f x 2 - 1 n 1 sin nx -n x n . n n 1 3 . If f x is defined on the interval 0 l and satisfies the Dirichlet conditions it can be represented by the cosine Fourier series as well as the sine Fourier series with the help of the above formulas . Both series on the interval 0 l give the values of f x at points of its continuity and the value f x0 0 f x0 - 0 at points of its discontinuity outside the interval 0 l these two series represent different functions. . Fourier series in complex form. The complex Fourier expansion of a function f x on an interval -l l has the form œ f x V CnéUn x n -œ where n r Cn -1 f x e nX dx n 0 1 2 . l 2l J-i The expressions i nX are called harmonics the coefficients Cn are complex amplitudes xn are wave numbers of the function f x and the set of all wave numbers wn is called the discrete spectrum of the function. The cosine Fourier expansion of f x on the interval 0 l corresponds to the extension of f x to the interval -l 0 as an even function f -x f x . The sine Fourier expansion of f x on 0 l corresponds to the extension of f x to the interval -l 0 as an odd function f -x -f x