Mathematical Tools for Physics
2—Infinite Series 53 Ifcaorbthen this is really not very different from the preceding case, whereaandbare zero. Ifa= 0this is F ...
2—Infinite Series 54 If you simply setc= 0in this equation you get Fz= Q 1 Q 20 2 π^2 0 ∫π/ 2 0 dθ [ 4 a^2 sin^2 θ ] 3 / 2 The ...
2—Infinite Series 55 This is an elementary integral. Letθ= (c/ 2 a) tanφ. ∫Λ 0 dθ [ c^2 + 4a^2 θ^2 ] 3 / 2 = ∫Λ′ 0 (c/ 2 a) sec^ ...
2—Infinite Series 56 Problems 2.1 If you borrow $200,000 to buy a house and will pay it back in monthly installments over 30 yea ...
2—Infinite Series 57 2.8 Determine the Taylor series forcoshxandsinhx. 2.9 Working strictly by hand,evaluate^7 √ 0. 999. Also √ ...
2—Infinite Series 58 2.14 Use series expansions to evaluate lim x→ 0 1 −cosx 1 −coshx and lim x→ 0 sinkx x 2.15 Evaluate using s ...
2—Infinite Series 59 2.20 Determine the double power series representation about(0,0)of 1 (1−x/a)(1−y/b) 2.21 Determine the doub ...
2—Infinite Series 60 2.25 The electric potential from one point charge iskq/r. For two point charges, you add the potentials of ...
2—Infinite Series 61 2.30 You know the power series representation for the exponential function, but now apply it in a slightly ...
2—Infinite Series 62 d ay 2.34 A massm 1 hangs from a string that is wrapped around a pulley of massM. As the massm 1 falls with ...
2—Infinite Series 63 2.38 A function is defined by the integral f(x) = ∫x 0 dt 1 −t^2 Expand the integrand with the binomial exp ...
2—Infinite Series 64 Iterate on this and compare it to the series expansion of the exact solution. Solve 0. 001 x^2 +x+ 1 = 0. 2 ...
Complex Algebra When the idea of negative numbers was broached a couple of thousand years ago, they were considered suspect, in ...
3—Complex Algebra 66 so(1,0)has this role. Finally, where does √ − 1 fit in? (0,1)(0,1) = (0. 0 − 1. 1 , 0 .1 + 1.0) = (− 1 ,0) ...
3—Complex Algebra 67 That was easy, what about the square root? A little more work: √ z=w=⇒z=w^2 Ifz=x+iyand the unknown isw=u+i ...
3—Complex Algebra 68 Complex Exponentials A function that is central to the analysis of differential equations and to untold oth ...
3—Complex Algebra 69 f(0) = 1 =A, f′(y) =−Asiny+Bcosy, f′(0) =−g(0) = 0 =B This determines thatf(y) = cosyand then Eq. ( 5 ) det ...
3—Complex Algebra 70 When you’re adding or subtracting complex numbers, the rectangular form is more convenient, but when you’re ...
3—Complex Algebra 71 Apply Eq. ( 8 ) for the addition of angles to the case thatθ=x+iy. cos(x+iy) = cosxcosiy−sinxsiniy= cosxcos ...
3—Complex Algebra 72 Examples Simplify these expressions, making sure that you can do all of these manipulations yourself. 3 − 4 ...
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