A change of basis matrix and integrals of power of cosine. (English) Zbl 1209.15008
Summary: This work is based on the idea of J. W. Rogers jun. [Am. Math. Mon. 104, No. 1, 20–26 (1997; Zbl 0955.15002)] who suggested to use the inverse of a change of basis matrix to find the antiderivative of a function. We use properties of the Chebyshev polynomial of the first kind to derive our change of basis matrix. We give the explicit entries for its inverse and apply it to achieve a formula for integrating the power of cosine.
MSC:
| 15A04 | Linear transformations, semilinear transformations |
| 15A09 | Theory of matrix inversion and generalized inverses |
| 26A06 | One-variable calculus |
Citations:
Zbl 0955.15002References:
| [1] | Benjamin, A. T.; Ericksen, L.; Jayawant, P.; Shattuck, M., Combinatorial trigonometry with Chebyshev polynomials, J. Statist. Plann. Inference, 140, 2157-2160 (2010) · Zbl 1230.05093 |
| [2] | Benjamin, A. T.; Walton, D., Counting on Chebyshev polynomials, Math. Mag., 82, 117-126 (2009) · Zbl 1223.33013 |
| [3] | Benjamin, A. T.; Walton, D., Combinatorially composing Chebyshev polynomials, J. Statist. Plann. Inference, 140, 2161-2167 (2010) · Zbl 1230.05085 |
| [4] | Graham, R. L.; Knuth, D. E.; Patashnik, O., Concrete Mathematics: A Foundation for Computer Science (1994), Addison-Wesley Professional: Addison-Wesley Professional New York · Zbl 0836.00001 |
| [5] | Rogers, J. W., Applications of linear algebra in calculus, Am. Math. Monthly, 104, 20-26 (1997) · Zbl 0955.15002 |
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