The derivative of a function f(x) is another function, defined in terms of a limiting expression: f(x + δx) f(x)

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1 Y. D. Chong (2016) MH2801: Complex Methos for the Sciences 1. Derivatives The erivative of a function f(x) is another function, efine in terms of a limiting expression: f (x) f (x) lim x δx 0 f(x + δx) f(x). (1) δx If the limiting expression give a unique an well-efine result within some omain of x, then we say that the erivative exists in that omain. We also say that f(x) is ifferentiable in that omain. It can be shown that a ifferentiable function is automatically continuous. (Try proving it!) For the purposes of imensional analysis, the erivative of a function f(x) has the units of the original function, ivie by the units of x. Secon-orer an higher-orer erivatives are efine by repeating the erivative proceure. Graphically, the erivative represents the slope of the graph of f(x), while the secon erivative represents the curvature. For example, the graph above has positive secon erivative, because it is upwar-curving. Derivatives obey several elementary composition rules: x [α f(x) + β g(x)] α f (x) + β g (x) (linearity) (2) x [f(x) g(x)] f(x) g (x) + f (x) g(x) (prouct rule) (3) x [f(g(x))] f (g(x)) g (x) (chain rule) (4) These can all be proven by irectly substituting into Eq. (1), an taking appropriate orers of limits. With the ai of these rules, we can prove various stanar results, such as the power rule for erivatives: [ x a ] ax a 1. x The linearity of the erivative operation has the important implication that erivatives commute with sums, i.e. you can move them to the left or right of summation signs. For example, we can use this to show that the exponential function is its own erivative: x [exp(x)] x n0 x n n! n0 x n x n! n1 x n 1 exp(x). (5) (n 1)! Derivatives also commute with limit expressions. For example, we can use this on the alternative efinition of the exponential function: x [exp(x)] x lim lim n n ( 1 + n) x n ( lim 1 + x ) n n x n ) n 1 (6) exp(x) ( 1 + x n 7

2 Y. D. Chong (2016) MH2801: Complex Methos for the Sciences 1.1 Taylor series A function is infinitely ifferentiable if all orers of erivatives are well-efine (i.e., first erivative, secon erivative, etc.). Not all functions behave this way: for example, f(x) x has a first erivative which is iscontinuous at x 0, which means that it has no well-efine secon erivative at that point. If a function is infinitely ifferentiable, then near any point x 0 it can be written out in a Taylor series: (x x 0 ) n [ ] n f(x) n! x n f (x 0 ) (7) n0 f(x 0 ) + (x x 0 ) f (x 0 ) + (x x 0) 2 f (x 0 ) + (8) 2 Here, the zeroth erivative refers to the function itself. The Taylor series can be erive by assuming that f(x) can be written out as a general polynomial involving terms of the form (x x 0 ) n ; from the efinition of the erivative, we can then obtain the polynomial coefficients. Because we are only assuming the valiity of the the polynomial series, there is no guarantee that the Taylor series converges to the true value for any value of x. For many functions, the Taylor series converges only if x x 0 is smaller than a certain amount Useful Taylor Series 1 1 x 1 + x + x2 + x 3 + for x < 1 (9) ln(1 x) x x2 2 x3 3 x4 for x < 1 (10) 4 sin(x) x x3 3! + x5 5! x7 7! + (11) sinh(x) x + x3 3! + x5 5! + x7 7! + (12) cos(x) 1 x2 2! + x4 4! x6 6! + (13) cosh(x) 1 + x2 2! + x4 4! + x6 6! + (14) Apart from the first two, the others are actually exact for all x. The sine an cosine functions an the hyperbolic functions are entire, which means that their Taylor series converge everywhere in the omain R. If you pick a large value of x, however, you may have to sum to a very high orer before the series converges to an accurate value. 1.2 Orinary ifferential equations A ifferential equation is an equation involving several ifferent erivatives of a function. For example, f κf(x) (15) x involves both f(x) an its first erivative. Specifically, this is an orinary ifferential equation, because f(x) is a function involving a single input number x. There is no single fixe proceure for solving ifferential equations. In some cases, we can guess the solution: for example, we might happen to know that the above ifferential equation has solutions of the form f(x) A exp(κx). (16) 8

3 Y. D. Chong (2016) MH2801: Complex Methos for the Sciences There are methos, such as Green s functions, for generating solutions to certain special classes of ifferential equation. But many ifferential equations simply cannot be solve analytically, an can only be solve numerically. When confronte with an orinary ifferential equation, typically the first thing you shoul look for is the orer of the highest erivative appearing in the equation. This is also calle the orer of the ifferential equation. If the equation has orer N, then its general solution contains N inepenent parameters, or constants of integration. Now, if you happen to be able to guess a solution to the equation, but that solution oes not contain N parameters, then you know that your solution isn t the most general one. For example, in Eq. (15), the orer is 1. We guesse the solution (16), which has one parameter A. So we know our work is one: there is no solution more general than the one we foun. A specific solution to a ifferential equation is a solution in which there are no inepenent parameters. This can be one by assigning numerical values to the inepenent parameters of the general solution. These numerical values are often specifie in terms of bounary conitions. For example, for a secon-orer ifferential equation, I might give you f(x 0) an f (x 0), an ask you to fin the corresponing specific solution. Alternatively, I coul give you f(x 0) an f(x 1), or any other combination of two conitions. For an orinary ifferential equation of orer N, we nee N bounary conitions to arrive at a specific solution. Example The following ifferential equation escribes a ampe harmonic oscillator: 2 x x + 2γ t2 t + ω2 0x(t) 0. (17) In this case, note that x(t) is the function, an t is the input variable. This is unlike our previous notation where x was the input variable; on t get confuse! Eq. (17) is obtaine from Newton s secon law, for an object moving in one imension subject to a amping force an a restoring force. Thus x(t) represents the position as a function of time. We will come back to this equation when stuying the ampe harmonic oscillator in greater etail. 1.3 Partial erivatives So far, we have focuse on functions which take a single input. Functions can also take multiple inputs; for instance, a function f(x, y) maps two input numbers, x an y, an outputs a number. In general, the inputs are allowe to vary inepenently of one another. The partial erivative of such a function is its erivative with respect to one of its inputs, keeping the other input fixe. For example, has partial erivatives f(x, y) sin(2x 3y 2 ) (18) x 2 cos(2x 3y2 ), y 6 cos(2x 3y2 ). (19) Change of variables We ve seen that single-variable functions obey a erivative composition rule x f( g(x) ) g (x) f ( g(x) ). (20) 9

4 Y. D. Chong (2016) MH2801: Complex Methos for the Sciences This composition rule has a important generalization for partial erivatives, which is relate to the physical concept of a change of coorinates. Suppose we have a function f(x, y) which takes two inputs x an y, an wish to express them using a ifferent coorinate system enote (say) u an v. In general, each coorinate in the ol system woul epen on both coorinates in the new system: Expresse in the new coorinates, the function is x x(u, v), y y(u, v). (21) F (u, v) f ( x(u, v), y(u, v) ). (22) It can then be shown that this transforme function s partial erivatives obey the composition rule u x x u + y y u (23) v x x v + y y v. (24) On the right-han sie of these equations, the partial erivatives are to be expresse in terms of the new cooriantes (u, v). For example, x x (25) xx(u,v), yy(u,v) The generalization of this rule to more than two inputs is straightforwar. For a function f(x 1,..., x N ), a change of coorinates x i x i (u 1,..., u N ) involves the composition F (u 1,..., u N ) f ( x 1 (u 1,..., u N ),... ), u i N j1 x j. (26) u i x j Example In two imensions, Cartesian an polar coorinates are relate by the formulas x r cos θ, y r sin θ. (27) If we have a function f(x, y), we can re-write it in polar coorinates as F (r, θ). The partial erivatives are relate by r x x r + y y r x θ x x θ + y y θ x cos θ + sin θ. (28) y r sin θ + r cos θ. (29) y This is often written in matrix form as: [ ] [ ] [ ] / r cos(θ) sin(θ) / x. (30) (1/r) / θ sin(θ) cos(θ) / y Partial ifferential equations A partial ifferential equation is a ifferential equation which involves multiple partial erivatives (as oppose to an orinary ifferential equation, which involves only erivatives with 10

5 Y. D. Chong (2016) MH2801: Complex Methos for the Sciences respect to a single variable). equation, An example of a partial ifferential equation is Laplace s 2 Φ x Φ y Φ 0, (31) z2 which escribes the electrostatic potential Φ(x, y, z) at position (x, y, z), in the absence of any electric charges. Partial ifferential equations are significantly harer to solve than orinary ifferential equations. For example, bounary conitions are more complicate to specify: whereas each bounary conition for an orinary ifferential equation consists of a single number (e.g., the value of f(x) at some point x x 0 ), each bounary conition for a partial ifferential equation consists of a function (e.g., the values of Φ(x, y, z) along some curve g(x, y, z) 0). 1.4 Exercises 1. Prove that x [xy ] yx y 1, for x R +, y / N, (32) starting from the previously-iscusse efinition of non-natural powers, in terms of the exponential an logarithm functions. 2. Consier f(x) tanh(αx). Sketch f(x) versus x, for two cases: (i) α 1 an (ii) α 1. Sketch the erivative function f (x) for the two cases, base on your sketches in part (a) (i.e., without evaluating the erivative irectly). Evaluate the erivative function, an verify that the result matches your sketches in part (b). 3. Prove geometrically that the erivatives of the sine an cosine functions are: sin(x) cos(x), x Hence, erive their series expansions. cos(x) sin(x). (33) x 4. For each of the following functions, erive the Taylor series aroun x 0: f(x) ln [α cos(x)], to the first 3 non-vanishing terms. f(x) cos [π exp(x)], to the first 4 non-vanishing terms. 1 f(x), to the first 4 non-vanishing terms. Keep track of the signs (i.e., 1 ± x ± versus ). 5. For each of the following functions, sketch the graph an state the omains over which the function is ifferentiable: f(x) sin(x) f(x) [tan(x)] 2 f(x) 1 1 x 2 6. Let v(x) be a vectorial function which takes an input x (a number), an gives an output value v that is a 2-component vector. The erivative of this vectorial function is efine in terms of the erivatives of each vector component: v(x) [ ] v1 (x) v 2 (x) v x [ v1 /x v 2 /x ]. (34) 11

6 Y. D. Chong (2016) MH2801: Complex Methos for the Sciences Now suppose v(x) obeys the vectorial ifferential equation v A v, where A x [ ] A11 A 12. (35) A 21 A 22 How many inepenent numbers o we nee to specify for the general solution? Next, let u be one of the eigenvectors of A, with eigenvalue λ: A u λ u. (36) Show that v(x) u e λx is a specific solution to the vectorial ifferential equation. Hence, fin the general solution. 7. Show that if a function is ifferentiable, then it is also continuous. 12