Question

Obtain the formal expansion of the function f defined by f(x)=lnx on (1≤x≤e^2π) is a series of orthonormal characteristic functions of SturmLiouville problem d/dx[x dy/dx]λ​/x y=0 y'(1) =0 y'(e^2π) =0 where

λ​=0

Answer 1

To formally expand the function f(x) = ln x, one needs to solve the given Sturm-Liouville problem to find orthonormal eigenfunctions and then use them to expand the logarithm function in terms of this function set.

Step-by-step explanation:

The question asks to expand the function f(x) = ln x into a series within the interval (1 ≤ x ≤ e^{2π}) based on orthonormal characteristic functions related to the Sturm-Liouville problem. In such problems, we often look for eigenfunctions and eigenvalues, which in this case, correspond to a differential equation with specific boundary conditions. The series expansion of a function into a set of orthonormal functions is akin to expressing a vector in terms of basis vectors in linear algebra. However, to answer this question, one would need to solve the Sturm-Liouville problem first to obtain the eigenfunctions and then project the logarithm function onto them.

Power series expansions are used to represent standard mathematical functions like trigonometric, logarithmic, and exponential functions. For logarithmic functions, using properties such as ln(xy) = ln x + ln y can facilitate the expansion process. Additionally, relationships like In (e^x) = x and e^{ln x} = x are used in manipulations involving exponents and logarithms.

The answer to this question involves using mathematical concepts like dimensional consistency, power series, and the properties of logarithms and exponents to construct an understanding of the required series expansion in terms of eigenfunctions found in the here given Sturm-Liouville problem.