How is this justified? ##e^{\cos(mx)} \cdot \cos(ny) \ne e^{\cos(mx + ny)}##.uaefame said:Homework Statement
The bessel function Jn(x) is defined by the integral
Jn(x)=1/(inπ)∫0πeixcosφcos(nφ)dφ
From this formula, find the first 3 terms of the asymptotic expansion of Jn(x) when x=n and n is a large positive integer.
Homework Equations
The Attempt at a Solution
I tried combining the cos and exp term together
Jn(x) = ∫-ππ ei(xcosφ+nφ)dφ
Mark44 said:How is this justified? ##e^{\cos(mx)} \cdot \cos(ny) \ne e^{\cos(mx + ny)}##.
The asymptotic expansion of Jn(x) is a mathematical series that approximates the Bessel function of the first kind, Jn(x), for large values of x. It is used to calculate the values of the Bessel function with high precision.
The asymptotic expansion of Jn(x) is calculated using the Laplace method, which involves expanding the Bessel function into a power series and then applying the method of steepest descent to determine the dominant term in the series.
The first 3 terms of the asymptotic expansion of Jn(x) are given by:
Jn(x) ~ (1/√2πx)cos(x-πn/2) + (1/8πx^2)sin(x-πn/2) - (1/8πx^3)cos(x-πn/2)
The accuracy of the asymptotic expansion of Jn(x) depends on the value of x and the number of terms included in the expansion. For large values of x, including more terms in the expansion can improve the accuracy of the approximation.
The asymptotic expansion of Jn(x) has many practical applications in physics, engineering, and other fields. It is commonly used in the analysis of wave phenomena, such as diffraction and scattering, and in solving differential equations that involve Bessel functions. It is also used in the study of heat transfer and electrical engineering problems.