Mathematical Tools for Physics
3—Complex Algebra 73 and other integerskjust keep repeating these three values. e^6 πi/^5 e^4 πi/^5 e^8 πi/^5 e^2 πi/^5 1 5 thro ...
3—Complex Algebra 74 3.5 Mapping When you apply a complex function to a region in the plane, it takes that region into another r ...
3—Complex Algebra 75 ...
3—Complex Algebra 76 Problems 3.1 Pick a pair of complex numbers and plot them in the plane. Compute their product and plot that ...
3—Complex Algebra 77 3.10 For arbitrary integern > 1 , compute the sum of all thenthroots of one. (When in doubt, try a coupl ...
3—Complex Algebra 78 3.21 What isii? Ans: If you’d like to check your result, typei∧iintoGoogle. 3.22 For what argument doessinθ ...
3—Complex Algebra 79 3.25 If the coefficients in a quadratic equation are real, show that ifzis a complex root of the equation t ...
3—Complex Algebra 80 3.31 Find the sum of the series ∞ ∑ 1 in n 3.32 Evaluate|cosz|^2. Evaluate|sinz|^2. 3.33 Evaluate √ 1 +i. E ...
3—Complex Algebra 81 3.37 Look at the vertical lines in thez-plane as mapped by Eq. ( 12 ). I drew the images of linesy=constant ...
3—Complex Algebra 82 3.45 Use Eq. ( 9 ) and look back at the development of Eq. (1.4) to find thesin−^1 andcos−^1 in terms of lo ...
Differential Equations The subject of ordinary differential equations encompasses such a large field that you can make a profess ...
4—Differential Equations 84 the spring is to a good approximation proportional to the distance that the mass has moved from its ...
4—Differential Equations 85 All you have to do is to recall that the derivative of an exponential is an exponential. det/dt=et. ...
4—Differential Equations 86 Solve forAandBto getA=−B=v 0 /(2iω 0 ). Then x(t) = v 0 2 iω 0 [ eiω^0 t−e−iω^0 t ] = v 0 ω 0 sinω 0 ...
4—Differential Equations 87 For small values oft, the first terms in the power series expansion of this result are x(t) = v 0 ω′ ...
4—Differential Equations 88 Find anyonesolution to the full inhomogeneous equation. Note that for step one you have to have all ...
4—Differential Equations 89 The homogeneous part of Eq. ( 11 ) has the solution found in Eq. ( 6 ) and the total is x(t) =xhom(t ...
4—Differential Equations 90 Everything in the solution is proportional toF 0 and that’s not surprising. I’ll leave it as an exe ...
4—Differential Equations 91 that the initial conditions are obeyed only approximately. The exponential terms have oscillations a ...
4—Differential Equations 92 andQare not too bad. Specifically this requires that(x−x 0 )P(x)and(x−x 0 )^2 Q(x)have no singularit ...
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