[MUSIC] Our next topic is derivation and integration of formal power series. In our previous lecture we already discussed taking the derivative of a formal power series A(q) Which was a naught + a1q + a2q squared + etc. Its derivative was defined as a1 + 2a2q + 3 It should be 2, 3a3q squared + etc.+ nanq to the power n- 1 + etc. Okay, and we know that taking the derivative is linear, The sum of the derivative is the derivative of the sum. And that, well, it was left as an exercise. But it is really easy, that, The derivative of the product can be computed by the Leibniz rule. So AB prime is A prime times B + A times B prime. Okay, and, Having this definition, we already can solve some very simple differential equations. For instance, question. Find all such power series which satisfy the differential equation. Find all formal power series F(q), satisfying the equation F prime = F The solution is pretty simple. Just compute the coefficients of this formal power series one by one. If F is f naught + f1q + f2q2+ etc., then F prime is f1 + 2f2q + 3f3 q2+ etc. And these two power series must be equal. So, So fn = (n +1)fn+1. For all n greater than or equal to 0. And in this case, you can find them one by one. And so you see that fn is nothing but f0 divided by n factorial. So if f0 = c, then our F(q) has the form c(1 +1 over 1 factorial q +1 over 2 factorial q2 + etc.+ 1 over n factorial qn + etc.). And in this series you probably recover the power series for the exponent. So exp(q) is 1 + 1 over 1 factorial q + 1 over 2q2+ 1 over 3 factorial q3 +etc. Then, F(q) is nothing but a multiple of the exponent. I'm sorry, there should be, of course this should be squared over here, not q to the third. Okay, so we have the power series for the exponent. And we'll use it further. Okay, but before we do that, let us define the operation, which is inverse to taking the derivative. This is integration of formal power series. So, Integration can also be defined formally by the same formulas as you have with polynomials. So the integral of A(q) is defined as a naught q + a1 over 2q2 + a2 over 3 q3 + etc.+ an over n+1 q n+1 +etc. Okay, and, You see that what happens if you start with some formal power series, then to compute its derivative, And then take its integral. Well, the first guess would be to say that you get what you started with, named A(q), but this is not completely true. Because you see that the integral of formal power series has no constant term by definition. So there's is no constant term here. This series starts with a naught q. So what you get by taking the integral is, The series A(q) without its constant term. So it is a1q +a2q2+ A3q3 + etc. And it is, A(q)- its constant term. Or to put it in a different way, A(q)- A(0). And this, if you wish, this can be created as the Newton-Leibniz formula. Newton-Leibniz formula is familiar to you from mathematical analysis. Further, we'll prove the chain rule, which tells us how to differentiate the composition of two functions. You already know the analog of the chain rule from mathematical analysis. And we will see that for formal power series, things are pretty much the same. [MUSIC]