# What is meant by "applying rotation in the space of $\gamma$" that makes the integral equals an infinite series?

In this paper, the author in proving the lemma

Let $$g(x) \geq 0$$ be defined for $$x \geq 0$$ and vanish for $$x > 1,$$ such that $$g(v_1^2 +\ldots + v_n^2)$$ is summable. Let $$f(w_1, \ldots, w_m) \geq 0$$ be summable. In the product space of the $$v$$'s and $$w$$'s let $$R$$ be a region such that $$\int_Rf(w_l, \ldots, w_m)g(v_1^2 +\ldots + v_n^2)\exp(\gamma_1v_1 +\ldots\gamma_n v_n)~dv dw = G(\gamma_1^2 +\ldots+\gamma_n^2)\tag 1.$$ Let $$w_0$$ be the region defined by the inequality $$v_1^2 + \ldots v_n^2 \geq k.\tag 2$$ Let $$\int_{R_0}f(w_l, \ldots, w_m)g(v_1^2 +\ldots + v_n^2)\exp(\gamma_1v_1 +\ldots\gamma_n v_n)~ dv dw = G_0(\gamma_1^2 +\ldots+\gamma_n^2)\tag 3.$$ Finally, let $$\int_Rf(w_l, \ldots, w_m)g(v_1^2 +\ldots + v_n^2)~ dv dw = \int_{R_0}f(w_l, \ldots, w_m)g(v_1^2 +\ldots + v_n^2)~ dv dw \tag 4.$$ Then $$G(x)\leq G(x_0)\tag 5$$ for all positive $$x.$$

first sets $$\gamma_1 = x$$ and other elements $$0$$ getting

$$G(x^2) = \int_Rf(w_l, \ldots, w_m)g(v_1^2 +\ldots + v_n^2)\exp(xv_1)~ dv dw .$$ He then claims this and conditions on $$f, ~g$$ imply $$G$$ is continuous. He then multiplies $$(1)$$ by $$\exp(-\sum \gamma_i^2)$$ and integrating both sides over $$0\leq a\leq \sum \gamma_i^2\leq b$$ to get $$K\int_a^b x^{\frac12(n-2)}e^{-x}G(x)~ dx = \int_R f(w_1, \ldots, w_m)g\left(\sum v_i^2\right)~ dv dw\int_{a\leq \sum \gamma_i^2\leq b} \exp\left(-\sum \gamma_i^2 + \sum \gamma_i v_i\right)~d\gamma.\tag 6$$ Then he writes

Applying a rotation in the space of the $$\gamma$$'s to the inner integral in the right-hand side of $$(6),$$ we obtain \begin{align}K\int_a^b x^{\frac12(n-2)}e^{-x}G(x)~ dx &= \int_R f(w_1, \ldots, w_m)g\left(\sum v_i^2\right)~ dv dw\int_{a\leq \sum x_i^2\leq b} \exp\left[-\sum x_i^2 + \left(\sum v_i^2\right)^{\frac12}x_1\right]dx\\ &= \sum_{h=0}^\infty c_h I_h(R)\end{align} where $$I_h(R) = \frac{1}{(2h)!}\int_R f(w_1, \ldots, w_m)g\left(\sum v_i^2\right)^h~ dv dw$$ and $$c_h = \int_{a\leq \sum x_i^2\leq b} x_1^{2h}\exp\left(-\sum x_i^2\right)~ d x.$$

This is where I couldn't comprehend what the author had done.

My questions are:

$$\bullet$$ What did author mean by "applying a rotation in the space of $$\gamma$$s"? How did that "rotation" enable him to go from $$(6)$$ to express the integral as the infinite sum of $$c_h I_h(R)?$$ How did the rotation turn $$\sum \gamma_i v_i$$ into $$\left(\sum v_i^2\right)^{\frac12}x_1?$$

$$\bullet$$ In $$(6),$$ why is there $$x^{\frac12(n-2)}?$$ How does it appear in the LHS?