Expansion of ex cos(x): A Comprehensive Guide
The function ex cos(x) combines two fundamental mathematical functions: the exponential function ex and the cosine function cos(x). An understanding of how to expand such a function is essential in various fields of mathematics and physics. Here, we will explore the process of finding the expansion of ex cos(x) through the application of Taylor and Maclaurin series.
Taylor Series Expansion
The Taylor series expansion for ex around x 0 is well-known:
[ e^x sum_{n0}^{infty} frac{x^n}{n!} ]The Taylor series expansion for cos(x) around x 0 is also widely used:
[ cos(x) sum_{n0}^{infty} frac{(-1)^n x^{2n}}{2n!} ]By multiplying these two series together, we can find the expansion of ex cos(x):
[ e^x cos(x) left( sum_{n0}^{infty} frac{x^n}{n!} right) left( sum_{m0}^{infty} frac{(-1)^m x^{2m}}{2m!} right) ]Multiplication of the Series
The product of two series can be expressed as:
[ e^x cos(x) sum_{n0}^{infty} sum_{m0}^{infty} frac{(-1)^m x^{n 2m}}{n!2m!} ]Rearranging the Series
To combine this series, we can let k n 2m. Thus, the series can be rewritten as:
[ e^x cos(x) sum_{k0}^{infty} left( sum_{m0}^{leftlfloor frac{k}{2} rightrfloor} frac{(-1)^m}{k-2m!2m!} right) x^k ]This series represents an expansion that converges for all x. For practical calculations, we can compute a few terms from the series expansions of ex and cos(x) directly and multiply them out to find the coefficients for specific powers of x.
Alternative Expansion Using Taylor-Mclaurin Series
We can also consider the function fx ex cos(x), which is differentiable as many times as desired. The Taylor-Mclaurin series expansion can be derived by finding the successive derivatives of the function fx and evaluating them at x 0.
[ e^x cos(x) f(0) Df(0)x frac{D^2f(0)}{2!}x^2 ldots frac{D^nf(0)}{n!}x^n ldots ]Alternatively, we can observe that for z x i y, with x and y being real numbers:
[ e^x Releft[ e^x i x right] Releft[ e^{i x} right] 1 - frac{x^2}{2!} frac{x^4}{4!} - ldots ]which can be further expanded as:
[ e^x 1 - frac{x}{6} - frac{x^3}{30} ldots ]Conclusion
In conclusion, the expansion of ex cos(x) is given by:
[ e^x cos(x) sum_{k0}^{infty} left( sum_{m0}^{leftlfloor frac{k}{2} rightrfloor} frac{(-1)^m}{k-2m!2m!} right) x^k ]This series converges for all x. For practical applications, we can compute a few terms of the series for ex and cos(x) and multiply them out to find the coefficients for specific powers of x.