Power Series Expansions for Trigonometric Functions via Solutions to Initial Value Problems

1991 ◽  
Vol 64 (4) ◽  
pp. 247
Author(s):  
A. P. Stone
Keyword(s):  
Power Series ◽  
Initial Value Problems ◽  
Series Expansions ◽  
Trigonometric Functions ◽  
Initial Value ◽  
Power Series Expansions

Superexponentials: A Generalization of Hyperbolic and Trigonometric Functions

2017 ◽  
Vol 3 (1) ◽  
pp. 7
Author(s):  
Alfonso F. Agnew ◽  
Brandon Gentile ◽  
John H. Mathews
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Initial Value Problem ◽  
Second Order ◽  
Series Expansions ◽  
Roots Of Unity ◽  
Trigonometric Functions ◽  
Initial Value ◽  
Second Order Differential Equations ◽  
Power Series Expansions

We construct and explore the properties of a generalization of hy- perbolic and trigonometric functions we cal l superexponentials. The general ization is based on the characteristic second-order differential equations (DE) these functions satisfy, and leads to functions satisfying analogous mth order equations and having many properties analogous to the usual hyperbolic and trigonometric functions. Roots of unity play a key role in providing the periodicity resulting in various properties. We also show how these functions solve the general initial value problem for the differential equations y(n) = y, and a look at the power series expansions reveal surprisingly simple patterns that clarify the properties of the superexponentials.

scholarly journals Parallel Software to Offset the Cost of Higher Precision

2021 ◽  
Vol 40 (2) ◽  
pp. 59-64
Author(s):  
Jan Verschelde
Keyword(s):  
Power Series ◽  
Parallel Algorithms ◽  
Series Expansions ◽  
Use Case ◽  
Double Precision ◽  
Algebraic Space ◽  
Space Curves ◽  
Computational Overhead ◽  
Power Series Expansions ◽  
The Cost

Hardware double precision is often insufficient to solve large scientific problems accurately. Computing in higher precision defined by software causes significant computational overhead. The application of parallel algorithms compensates for this overhead. Newton's method to develop power series expansions of algebraic space curves is the use case for this application.

scholarly journals CONGRUENCES AMONG POWER SERIES COEFFICIENTS OF MODULAR FORMS

2013 ◽  
Vol 09 (06) ◽  
pp. 1447-1474
Author(s):  
RICHARD MOY
Keyword(s):  
Power Series ◽  
Modular Forms ◽  
Series Expansions ◽  
Modular Functions ◽  
Apéry Numbers ◽  
Congruence Relations ◽  
Power Series Expansions

Many authors have investigated the congruence relations among the coefficients of power series expansions of modular forms f in modular functions t. In a recent paper, R. Osburn and B. Sahu examine several power series expansions and prove that the coefficients exhibit congruence relations similar to the congruences satisfied by the Apéry numbers associated with the irrationality of ζ(3). We show that many of the examples of Osburn and Sahu are members of infinite families.

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