Question

Please solve 6.26

(E) \( 6.26\left(\right. \) a ) Obtain three terms of the asymptotic expansion of \( \int_{0}^{n / 2} e^{-x \tan ^{2}} d \theta \) as \( x \rightarrow \infty \). (b) Find the leading behavior of \( \i

Answer 1

To obtain the asymptotic expansion, we can expand the integrand in powers of x and integrate each term.

Step-by-step explanation:

(a) Obtain three terms of the asymptotic expansion of ∫0^(n / 2) e-x∧tan2 dθ as x → ∞.

To obtain the asymptotic expansion, we can expand the integrand in powers of x and integrate each term. The first three terms of the expansion are:

e-x∧tan2 ≈ 1 - x∧tan2 + ½x2-tan4 + O(3)

(b) The leading behavior of ∫0^(n / 2) e-x∧tan2 dθ as x → ∞ is dominated by the first term in the asymptotic expansion. So, the leading behavior is 1.