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ISBN 3-211-82913-X. Mainardi, F. (2010). Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. Imperial College Press. ISBN 978-1-84816-329-4. Archived from the original on 2012-05-19. Tarasov, V.E. (2010). Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and ...
where is the Euler–Mascheroni constant which equals the value of a number of definite integrals. Finally, a well known result, ∫ 0 2 π e i ( m − n ) ϕ d ϕ = 2 π δ m , n for m , n ∈ Z {\displaystyle \int _{0}^{2\pi }e^{i(m-n)\phi }d\phi =2\pi \delta _{m,n}\qquad {\text{for }}m,n\in \mathbb {Z} } where δ m , n {\displaystyle \delta ...
A different technique, which goes back to Laplace (1812), [3] is the following. Let = =. Since the limits on s as y → ±∞ depend on the sign of x, it simplifies the calculation to use the fact that e −x 2 is an even function, and, therefore, the integral over all real numbers is just twice the integral from zero to infinity.
In the first case the line y = mx + n is an oblique asymptote of ƒ(x) when x tends to +∞, and in the second case the line y = mx + n is an oblique asymptote of ƒ(x) when x tends to −∞. An example is ƒ(x) = x + 1/x, which has the oblique asymptote y = x (that is m = 1, n = 0) as seen in the limits
The Riemann-Stieltjes integral is the area of the projection of this fence onto the f(x)-g(x) plane — in effect, its "shadow". The slope of g(x) weights the area of the projection. The values of x for which g(x) has the steepest slope g'(x) correspond to regions of the fence with the greater projection and thereby carry the most weight in the ...
Fejér's theorem states that the above sequence of partial sums converge uniformly to ƒ. This implies much better convergence properties If ƒ is continuous at t then the Fourier series of ƒ is summable at t to ƒ(t). If ƒ is continuous, its Fourier series is uniformly summable (i.e. () converges uniformly to ƒ).
In particular, if Y is a Banach space, then C(X, Y) is itself a Banach space under the uniform norm. The uniform limit theorem also holds if continuity is replaced by uniform continuity. That is, if X and Y are metric spaces and ƒ n : X → Y is a sequence of uniformly continuous functions converging uniformly to a function ƒ, then ƒ must be ...
In this case, an element x of the domain is represented by an interval of the x-axis, and the corresponding value of the function, f(x), is represented by a rectangle whose base is the interval corresponding to x and whose height is f(x) (possibly negative, in which case the bar extends below the x-axis).