Derivatives of the Trigonometric Functions

2020 ◽  
Vol 14 ◽  

This paper discusses the approximations with the local basis of the second level and the sixth order. We call it the approximation of the second level because in addition to the function values in the grid nodes it uses the values of the function, and the first and the second derivatives of the function. Here the polynomial approximations and the non-polynomial approximations of a special form are discussed. The non-polynomial approximation has the properties of polynomial and trigonometric functions. The approximations are twice continuously differentiable. Approximation theorems are given. These approximations use the values of the function at the nodes, the values of the first and the second derivatives of the function at the nodes, and the local basis splines. These basis splines are used for constructing variational-difference schemes for solving boundary value problems for differential equations. Numerical examples are given


Axioms ◽  
2019 ◽  
Vol 8 (4) ◽  
pp. 138 ◽  
Author(s):  
Giuseppe Dattoli ◽  
Silvia Licciardi ◽  
Rosa Maria Pidatella ◽  
Elio Sabia

Elementary problems as the evaluation of repeated derivatives of ordinary transcendent functions can usefully be treated with the use of special polynomials and of a formalism borrowed from combinatorial analysis. Motivated by previous researches in this field, we review the results obtained by other authors and develop a complementary point of view for the repeated derivatives of sec ( . ) , tan ( . ) and for their hyperbolic counterparts.


1975 ◽  
Vol 68 (7) ◽  
pp. 572-573
Author(s):  
R. S. Luthar

Here's a classic topic from analysis treated in an interesting new way with skillful application of identities and differentiation techniques.


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