CAS PROJECT. sec, tan, arcsin. (a) Euler numbers. The Maclaurin series E22 E44 (21) sec

Chapter 15, Problem 15.1.87

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CAS PROJECT. sec, tan, arcsin. (a) Euler numbers. The Maclaurin series

(21) \(\sec z=E_{0}-\frac{E_{2}}{2 !} z^{2}+\frac{E_{4}}{4 !} z^{4}-+\cdots\)

defines the Euler numbers \(E_{2 n}\). Show that \(E_{0}=1\), \(E_{2}=-1, E_{4}=5, E_{6}=-61\). Write a program that computes the \(E_{2 n}\) from the coefficient formula in (1) or extracts them as a list from the series. (For tables see Ref. [GR1], p. 810, listed in App. 1.)

(b) Bernoulli numbers. The Maclaurin series

(22) \(\frac{z}{e^{z}-1}=1+B_{1} z+\frac{B_{2}}{2 !} z^{2}+\frac{B_{3}}{3 !} z^{3}+\cdots\)

defines the Bernoulli numbers \(B_{n}\). Using undetermined coefficients, show that

\(B_{1} =-\frac{1}{2}, \quad B_{2}=\frac{I}{6} \quad B_{3}=0\)

(23)

\(B_{4} =-\frac{1}{30}, \quad B_{5}=0, \quad B_{6}=\frac{1}{42}, \cdots\)

Write a program for computing \(B_{n^{*}}\).

(c) Tangent. Using (1), (2), Sec. 13.6, and (22), show that tan z has the following Maclaurin series and calculate from it a table of \(B_{0}, \cdots, B_{20}\):

(24) \(\tan z =\frac{2 i}{e^{2 i z}-1}-\frac{4 i}{e^{4 i z}-1}-i\)

\(=\sum_{n=1}^{\infty}(-1)^{n-1} \frac{2^{2 n}\left(2^{2 n}-1\right)}{(2 n) !} B_{2 n} z^{2 n-1}\)

Text Transcription:

sec z = E_{0} - E_2 / 2! z^2 + E_4 / 4! z^4 - + cdots

E_2n

E_0 = 1

E_2 = -1, E_4 = 5, E_6 = -61

z / e^z - 1 = 1 + B_1 z + B_2 / 2! z^2 + B_3 / 3! z^3 + cdots

B_n

B_1 = - 1 / 2,     B_2 = 1 / 6     B_3 = 0

B_4 = - 1 / 30,   B_5 = 0,     B_6 = 1 / 42, cdots

B_n*

B_0, cdots, B_20

tan z = 2i / e^{2iz} - 1} - 4i / e^{4iz} - 1} - i

= sum_{n = 1}^{infty}(-1)^{n - 1} 2^2n (2^{2n} - 1) / (2n)! B_2n z^2n - 1