[MUSIC] >> Let us study the function t. So I would like to analyze. To functional equation for (z+1) and. For z- 1 over tau. Let us start by the first equation. By the definition of the function psi. Go ahead. E to the power i (z +1) to the square times tau. Theta tau z tau + tau. Because we calculate our function for z+ 1. With Jacobi theta function is quasi periodic, with index one half. We open the square. E to the power -pi i. Lambda is = 1. Lambda squared tau + 2 lambda, z lambda +1, theta, tau, Z, tau, and we have to add -1 because lambda here is =1, z=z tau. So we add -1 here. So now let us analyze the product. We can make the following simplification. The first one. The second one. So the result. This is- e to the power p i. Z squared tau times theta tao z tau. But this is -c and z. So we proved that our function [INAUDIBLE] satisfies the same functional equation as the data series in -1 tau. The second equation. Psi in z -1 over tau. By definition, this e to the power pi i, we see the definition on the screen. That -1 over tau to the square tau times data tau z tau -1. This is (= -1). Because here, mu = -1. E to the power p i z to the square tau, theta tau z tau. This is how our function psi times the rest e to the power pi i (-2z over tau +- over tau to the square Tau). What do we have here? Tau prime is equal to- over tau. And we can to write this as follows. -1 psi tau z. Here, we have e to the power -pi i -1 over tau +2 z. So you see that here. Tau prime is equal to -tau. It means lambda is =1. And here, we have 2. This is exactly the equation of the Jacobi theta of the Jacobi theta series. So we're really proved that our function psi satisfies the same relation as the Jacobi theta series, is -1 over tau z. They have the same 0. So, these function are the same up to a constant tell. Now, we have to calculate this function. [SOUND] So we proved then Jacobi theta series 1 over tau z is equal up to a constant depending on tau. Of function psi. The function psi is e to the power pi i z to the square tau times theta, (tau, z, tau). To find the coefficient c(t) I would like to find d over dz from the left hand side. And from the right hand side, and I would like to relate it for z=0. So this, Transformation gives us the following. The left hand side we get 2 pi i the Jacobi theta series in -1 over tau to the power 3. Because we analyzed. Maybe I'll write it here. To proved that D over D z theta comma zed. At z = 0 is=2 pi i the cube of the Dedekind theta function. This is our Jacobi Dedekind relation. Now, let us calculate the right hand side. T tau. If the find d z and that put z =0 here will get 0 and we get additional coefficient tau from this place. So we get 2 pi i. Tau. The Dedekind theta function to cube. Now, simplify. So, we can use the functional equation of the Dedekind theta function And this is square root from tau over e. We add this function into the Dedekind function, cubed. Theta tau cube. So now, we can calculate c t and we get C(tau) =- square root tau over i. Exactly, this vector with half in the formulation of the functional equation of Jacobi theta series for the evolution s. But for the translation t and evolution s generates a full group. So, now we can. Formulate the following corelation. Jacobi theta series in a (tau + b over c tau + d, z over c tau + d) = the cube of the multiplied of the system of Jacobi theta series of (c tau + d). The power 1/2 e to the power, pi i c z to the squared c tau + d, theta function Zed. So we have the cube of the multiplied system. Because, we have the cube of the multiplied system in the both modular equation. This is v ea t cubed. This is the eta cube. Certainly, this follows from this relation modulus effect that will prove that this is a modular form. So we see this Jacobi theta series is the Jacobi form. Of weight. 1/2 and index 1/2 with respect, with respect to the Jacobi group. With the character v h. The binary character of. The Heisenberg rule and multiply assistant. V at q of the full modular group. It takes value in the group of rules of order, 8 from 1. This is not a character, this is multiplier system and vh is a character. So this result is very, very important for our series. Let me write it once more in the following. Weight. This is Jacobi theta series of weight 1/2 and index 1/2, with the following multiplied system. The only fact to check is the fact that our modular form is holomorphic. [SOUND]