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pset1

# pset1 - The energy of an electron at speed v in special...

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Physics (PHZ) 3113 Mathematical Physics Florida Atlantic University Fall, 2010 Problem Set I Due: Tuesday, 31 August 2010 1. (Problems 1.10.5, 9 and 15, p. 22) Find the interval of convergence of each of the following power series. Be sure to investigate the endpoints of the interval in each case. S (a) = X n =1 x n ( n !) 2 , S (b) = X n =1 ( - 1) n n 3 x n and S (c) = X n =1 ( x - 2) n 3 n . 2. (Problems 1.13.9, 13 and 17, p. 32) Expand each of the following functions in a Maclaurin series (a Taylor series about x = 0). Write the series as an infinite sum and calculate the first few (non-zero) terms explicitly. f (a) ( x ) = 1 + x 1 - x , f (b) ( x ) = Z x 0 e - t 2 d t and f (c) ( x ) = ln 1 + x 1 - x . 3. (Problems 1.13.26, 31 and 35, p. 32) Find the first few terms of the Maclaurin series for each of the following functions and check your results by computer. f (a) ( x ) = 1 cos x , f (b) ( x ) = cos(e x - 1) and f (c) ( x ) = x sin x . 4. (Problems 1.15.7, 12 and 23c, pp. 40–41) Use Maclaurin series to evaluate the following limits. y (a) = d 8 d x 8 ( x 6 tan 2 x ) x =0 , y (b) = lim x 0 tan x - x x 3 and y (c) = lim x 0 csc 2 x - 1 x 2 . 5. (Problem 1.16.28, p. 42)

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Unformatted text preview: The energy of an electron at speed v in special relativity theory is E = mc 2 (1-v 2 /c 2 )-1 / 2 , where m is the electron mass, and c is the speed of light. The factor mc 2 is called the rest mass energy (the energy when v = 0). Find the ﬁrst three terms of the series expansion of E in v . What is the second term in the series? 6. (Problem 1.16.33, p. 43) If you are at the top of a tower of height h above the surface of the earth, show that the distance you can see along the surface of the earth is approximately s = √ 2 Rh , where R is the radius of the earth. (See the ﬁgure and the hints in the book.) Mathematical Physics, Fall, 2010 Problem Set I, Page 2 7. (Problem 4.12.16, p. 237) In kinetic theory, we have to evaluate integrals of the form I ( n ) = Z ∞ t n e-at 2 d t. Given that I (0) = p π/ 4 a , evaluate I for all integers n ....
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