[Calculation 9] Simple Corollaries from Gauss and Bailey Formula


Corollary 1. For $\frac{1}{2} < z < 2$, $$ {}_2 F_1 \left(\frac{1}{2}, \frac{1}{2};1;1-\frac{1}{z}\right) = \sqrt{z} {}_2 F_1 \left(\frac{1}{2}, \frac{1}{2};1;1-z\right). $$   Proof. We recall Bailey’s Formula ((i) in [2]) for $w\in\mathbb R$: \begin{equation}\tag{1} (1-w)^{-a} {}_2 F_1 \left( a,b;c; – \frac{w}{1-w}\right) … Continue reading

[Calculation 8] Gauss’s Quadratic Trasformation


Theorem. (Gauss’s Quadratic Transformation) \begin{equation}\tag{1} (1+z)^{-2a} {}_2 F_1 \left(a,b; 2b; \frac{4z}{(1+z)^2}\right) = {}_2 F_1 \left(a, 1+\frac{1}{2}-b;b+\frac{1}{2};z^2\right). \end{equation}   Proof. The proof is almost similar to that of [2]. Note that the left hand side of (1) can be expanded in … Continue reading

[Calculation 6] Kummer’s Theorem


Theorem. (Kummer’s Theorem) $$ {}_2 F_1 (a,b;1+a-b;-1) = \frac{\Gamma(1+a-b)\Gamma\left(1+\frac{1}{2}a\right)}{\Gamma(1+a)\Gamma\left(1+\frac{1}{2}a-b\right)} $$   To prove Kummer’s theorem, we introduce the following lemma, which is called Kummer’s quadratic transformation:   Lemma. (Kummer’s Quadratic Transformation) \begin{equation}\tag{1} {}_2 F_1 (a,b;1+a-b;z) = (1-z)^{-a} {}_2 F_1 \left( … Continue reading

[M1 Seminar II] Week 8 : Lorentz Invariance of the Wave Equations

Now we will introduce some lemmas for the proof of the global existence theorem for the nonlinear wave equations with quadratic nonlinearities. The exsistence theorem which was proven before gives $n >5$ for the quadratic nonlinearities. But in fact this … Continue reading