∫_(Σ_(1≤i≤n) x_i ^2 ≤1) (Σ_(1≤i≤n) x_i ^2 )^m (Σ_(1≤i≤n) a_i x_i )^(2k) Π_(1≤i≤n) dx


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Question Number 215540 by MrGaster last updated on 10/Jan/25

$\int_{\sum_{1 \leq i \leq n} x_{i}^{2} \leq 1} {(\sum_{1 \leq i \leq n} x_{i}^{2})}^{m} {(\sum_{1 \leq i \leq n} a_{i} x_{i})}^{2 k} \prod_{1 \leq i \leq n} {d x}_{i}$
	Answered by MrGaster last updated on 10/Jan/25

	$\int_{0}^{1} r^{2 m} d r \int_{\sum_{1 \leq i \leq x}} x_{i}^{2} = r^{2} {(\sum_{1 \leq i \leq n} a_{i} x_{i})}^{2 k} d s$ $\int_{0}^{1} r^{2 m + n + 1} d r \int_{\sum_{1 \leq i \leq n} x_{i}^{2} = 1} {(\sum_{1 \leq i \leq n} a_{i} x_{i})}^{2 k} d s$ $lf n = 2 v + 3 \in 2 N + 3, In order to avoid the combinationf$ $o n and v below the score .$ $I = \frac{1}{2 m + n + 2} \int_{\sum_{1 \leq i \leq n} x_{i}^{2} = 1} {(\sum_{1 \leq i \leq n} a_{i} x_{i})}^{2 k} d s$ $= \frac{1}{2 m + n + 2} \frac{2^{v + 2} π^{v + 1}}{(2 k + 1)! (2 k + 3) \dots (2 k + 2 v + 1)} Δ^{k} {(\sum_{1 \leq i \leq n} a_{i} x_{i})}^{2 k}$ $Δ^{k} {(\sum_{1 \leq i \leq 2 v + 3} a_{i} x_{i})}^{2 k} = {[\frac{\partial^{2}}{\partial x_{1}^{2}} + \frac{\partial^{2}}{\partial x_{2}^{2}} + \dots + \frac{\partial^{2}}{\partial x_{n}^{2}}]}^{k} e^{\frac{1}{a_{1}} \sum_{2 \leq i \leq n} a_{i} x_{i} \frac{\partial}{\partial x_{i}}} (a_{1} x_{1})$ $= {[\frac{\partial^{2}}{\partial x_{1}^{2}} + \frac{\partial^{2}}{\partial x_{2}^{2}} + \dots + \frac{\partial^{2}}{\partial x_{n}^{2}}]}^{k} e^{\frac{1}{a_{1}} \sum_{2 \leq i \leq n} a_{i} x_{i} \frac{\partial}{\partial x_{1}}} {(a_{1} x_{1})}^{2 k}$ $= {[\frac{\partial^{2}}{\partial x_{1}^{2}} + \frac{\partial^{2}}{\partial x_{2}^{2}} + \dots + \frac{\partial^{2}}{\partial x_{n}^{2}}]}^{k} e^{\frac{1}{a} \sum_{2 \leq i \leq n} a_{i} x_{i} \frac{\partial}{\partial x_{1}^{2 k}} x_{1}^{2 k}}$ $= e^{\frac{1}{a} \sum_{2 \leq i \leq n} a_{i} x_{i} \frac{\partial}{\partial x_{i}}} {(\sum_{1 \leq i \leq n} a_{k}^{2})}^{k} \frac{\partial^{2 k}}{\partial x_{1}^{2 k}} x_{1}^{2 k}$ $= (\sum_{1 \leq i \leq n} a_{i}^{2}) (2 k)!$ $I = \frac{2^{v + 2} π^{v + 1} {(\sum_{1 \leq i \leq n} a_{i}^{2})}^{k}}{(2 m + n + 2) (2 k + 1) (2 k + 3) \dots (2 k + 2 v + 1)}$ $lf n = 2 v + 2 \in 2 N + 2$ $I = \frac{1}{2 m + n + 2} \int_{\sum_{1 \leq i \leq n} x_{i}^{2} = 1} {(\sum_{1 \leq i \leq n} a_{i} + x_{i})}^{2 k} d s = \frac{1}{2 m + n + 2} \frac{2 π^{v + 1}}{4^{k} k! (v + k)} Δ^{k} {(\sum_{1 \leq i \leq n} a_{i} x_{i})}^{2 k}$ $= \frac{2 π^{v + 1} (2 k)! {(\sum_{1 \leq i \leq n} a_{i}^{2})}^{k}}{(2 m + n + 2) 4^{k} k! (v + k)!}$
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