Mathematical Tools for Physics
2—Infinite Series 33 This shows the terms of the series for the sine as stated. Does this show that the series converges? If it ...
2—Infinite Series 34 The second is an example of a Fourier series. Still another is the Frobenius series, useful in solving diff ...
2—Infinite Series 35 As long asx < 1 this is precisely set up for the comparison test using ∑ nuKx nas the series that domina ...
2—Infinite Series 36 The only difference between the infinite series on the left and on the right is one term, so either everyth ...
2—Infinite Series 37 For largen, the numerator is essentiallyn^3 and the denominator is essentially 5 n^5 , so for largenthis se ...
2—Infinite Series 38 2.4 Series of Series When you have a function whose power series you need, there are sometimes easier ways ...
2—Infinite Series 39 2.5 Power series, two variables The idea of a power series can be extended to more than one variable. One w ...
2—Infinite Series 40 Differentiate thismtimes with respect toxandntimes with respect toy, then setx=aandy=b. Only one term survi ...
2—Infinite Series 41 2.6 Stirling’s Approximation The Gamma function for positive integers is a factorial. A clever use of infin ...
2—Infinite Series 42 At the lower limit of the integral,t= 0, this integrand ise−n/^2 , so ifnis even moderately large then exte ...
2—Infinite Series 43 and the fractions of the time that you get each pair are respectively a^2 ba ab b^2 This says that the frac ...
2—Infinite Series 44 The original function is a maximum when this denominator is a minimum. When the numbersN andkare big, you c ...
2—Infinite Series 45 2.7 Useful Tricks There are a variety of ways to manipulate series, and while some of them are simple they ...
2—Infinite Series 46 If I multiply this byxI get 2 x+ 3x^2 + 4x^3 + 5x^4 +···and that starts to look like a derivative. xf(x) = ...
2—Infinite Series 47 Call the coordinate along the width of the slity, where 0 < y < a. I want to find the total light wav ...
2—Infinite Series 48 It’s the variation with angle that I want to look at. The intensity of the wave, the power per area, is pro ...
2—Infinite Series 49 Remember,ka/ 2 is big! This means that it makes sense to keep only one term of the sine expansion forsinθ i ...
2—Infinite Series 50 2.9 Checking Results When you solve any problem, or at least think that you’ve solved it, you’re not done. ...
2—Infinite Series 51 m 1 ax M m 2 Two masses are attached by a string of negligible mass and that is wrapped around a pulley of ...
2—Infinite Series 52 OR If there is no friction,μk= 0, thenm 1 plays no role in this result but if it is big then you know that ...
«
1
2
3
4
5
6
7
8
9
10
»
Free download pdf