Problem 1.2. Setting z = r € R and applying the saddle-point method, derive the asymptotic expansion for the real solutions of (1.1), for a >0.
Permutations and Combinations
If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.
Counting Theory
The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.
Please answer 1.2 and 1.3
![1. ASYMPTOTIC EXPANSIONS FOR SOLUTIONS OF IInd ORDER ODE
Consider the real solutions of the second-order ODE
(1.1)
f" = xf, rER.
Problem 1.1. Interpreting (1.1) as the real restriction of the compler ODE
d'u
dz?
= 2• u(2), zE C,
derive an integral representation for the solution u(2). Explicitly describe
the conditions satisfied by the contour of integration for this representation.
Problem 1.2. Setting z = r €R and applying the saddle-point method,
derive the asymptotic expansion for the real solutions of (1.1), for z >0.
Problem 1.3. Setting z = r €R and applying the stationary-phase method,
derive the asymptotic erpansion for the real solutions of (1.1), for a <0.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9ab6aba4-08ae-4aee-8989-a831d8286f98%2Ff936d7fb-f3ca-409d-a9c6-1cc08cff0aa5%2Fawngt28_processed.jpeg&w=3840&q=75)
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