Consider the functions C: R → R and S: R → R defined by 00 Σ Σ n=0 n=0 Define a constant to be 2x where x is the first positive zero of C(x) Show that C(x) = n 2n == (-1)"x² (2n)! and S(x) C() = 0, s() = 1 C(TT) 1, S(T) = 0 C(2T) = 1, S(2T) = 0 = (-1)" x ²" + (2n+1)!
Consider the functions C: R → R and S: R → R defined by 00 Σ Σ n=0 n=0 Define a constant to be 2x where x is the first positive zero of C(x) Show that C(x) = n 2n == (-1)"x² (2n)! and S(x) C() = 0, s() = 1 C(TT) 1, S(T) = 0 C(2T) = 1, S(2T) = 0 = (-1)" x ²" + (2n+1)!
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Consider the functions \( C: \mathbb{R} \to \mathbb{R} \) and \( S: \mathbb{R} \to \mathbb{R} \) defined by
\[
C(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} \quad \text{and} \quad S(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}
\]
Define a constant \(\pi\) to be \(2x_0\), where \(x_0\) is the first positive zero of \(C(x)\). Show that
\[
C\left(\frac{\pi}{2}\right) = 0, \quad S\left(\frac{\pi}{2}\right) = 1
\]
\[
C(\pi) = -1, \quad S(\pi) = 0
\]
\[
C(2\pi) = 1, \quad S(2\pi) = 0
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F37e8ed93-7bef-4409-89ed-52264f64a27e%2Fb0e8776b-63cf-495b-ab37-50cf9052eded%2Foet5g6q_processed.png&w=3840&q=75)
Transcribed Image Text:Consider the functions \( C: \mathbb{R} \to \mathbb{R} \) and \( S: \mathbb{R} \to \mathbb{R} \) defined by
\[
C(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} \quad \text{and} \quad S(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}
\]
Define a constant \(\pi\) to be \(2x_0\), where \(x_0\) is the first positive zero of \(C(x)\). Show that
\[
C\left(\frac{\pi}{2}\right) = 0, \quad S\left(\frac{\pi}{2}\right) = 1
\]
\[
C(\pi) = -1, \quad S(\pi) = 0
\]
\[
C(2\pi) = 1, \quad S(2\pi) = 0
\]
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