Mathematical Tools for Physics - Department of Physics - University

3—Complex Algebra 54 Solve the two equations foruandv. The result is 1 z = x−iy x^2 +y^2 (3.3) See problem3.3. At least it’s obv ...

3—Complex Algebra 55 3.3 Applications of Euler’s Formula When you are adding or subtracting complex numbers, the rectangular for ...

3—Complex Algebra 56 This polar form shows a geometric interpretation for the periodicity of the exponential.ei(θ+2π)= eiθ=ei(θ+ ...

3—Complex Algebra 57 andkis any integer.k= 0, 1 , 2 give 11 /^3 = 1, e^2 πi/^3 = cos(2π/3) +isin(2π/3), =− 1 2 +i √ 3 2 e^4 πi/^ ...

3—Complex Algebra 58 halfway alonga, turning left, then going the distance|a|/ 2. Now write out the two complex number P 1 −P 3 ...

3—Complex Algebra 59 3.7 Mapping When you apply a complex function to a region in the plane, it takes that region into another r ...

3—Complex Algebra 60 Exercises 1 Express in the forma+ib:(3−i)^2 , (2− 3 i)(3 + 4i). Draw the geometric representation for each ...

3—Complex Algebra 61 Problems 3.1 Pick a pair of complex numbers and plot them in the plane. Compute their product and plot that ...

3—Complex Algebra 62 3.15 Repeat the previous problem, but for the set of points such that thedifferenceof the distances from tw ...

3—Complex Algebra 63 cos ( kr 0 −ωt ) + cos ( k(r 0 −dsinθ)−ωt ) + cos ( k(r 0 − 2 dsinθ)−ωt ) + ...+ cos ( k(r 0 −Ndsinθ)−ωt ) ...

3—Complex Algebra 64 3.31 Find the sum of the series ∞ ∑ 1 in n Ans:iπ/ 4 −^12 ln 2 3.32 Evaluate|cosz|^2. Evaluate|sinz|^2. 3.3 ...

3—Complex Algebra 65 3.40 Evaluateziwherezis an arbitrary complex number,z=x+iy=reiθ. 3.41 What is the image of the domain−∞< ...

3—Complex Algebra 66 3.52 Confirm the plot ofln(1+iy)following Eq. (3.15). Also do the corresponding plots forln(10+iy) andln(10 ...

Differential Equations . The subject of ordinary differential equations encompasses such a large field that you can make a profe ...

4—Differential Equations 68 If there’s friction (and there’salwaysfriction), the force has another term. Now how do you describe ...

4—Differential Equations 69 Push this to the extreme case where the damping vanishes: b= 0. Thenα 1 =i √ k/mand α 2 =−i √ k/m. D ...

4—Differential Equations 70 Does it have the right size as well as the right sign? It is−v 0 γt^2 =−v 0 (b/ 2 m)t^2. But that’s ...

4—Differential Equations 71 as a constant and an exponential, it’s easy to verify that you add the results that you get for the ...

4—Differential Equations 72 Interchangeα 1 andα 2 to getB. The final result is x(t) = F 0 α 1 −α 2 ( α 2 (mβ^2 −bβ)−kβ ) eα^1 t− ...

4—Differential Equations 73 At timet= 0this is still zero even with the approximations. That’s comforting, but if it hadn’t happ ...