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## How to display following restrict of features?

Permit $ zeta in C_ {c} ^ { infty} (B (0,1)) $. Then as $ M rightarrow infty $

$$ M ^ {n-1} N int _ { Omega} zeta (M x) e ^ {- p N x_ {n}} dx rightarrow frac {1} {p} int _ { mathbb { R} ^ {n-1}} zeta left (x ^ { prime}, 0 privilege) dx ^ { prime} $$

Where $ (x_1, cdots, x_ {n-1}, x_n) = (x ‘, x_n) $, $ 1 and $ N = N (M) $ in order that $ M / N to $ 0 how $ M to infty $.

My try: With Change of variable we’ve $ (x ‘, x_n) = left ( frac {x’} {M}, frac {x_n} {M ^ 2} privilege) $. Jacobi = $ frac {1} {M ^ {n + 1}} $

commence {align *} M ^ {n-1} N int _ { Omega} zeta (M x) e ^ {- p N x_ {n}} d x & = M ^ {n-1} N int_ { Omega} zeta (x ‘, frac {x_n} {M}) e ^ {- pN frac {x_n} {M ^ 2}} frac {1} {M ^ {n + 1}} dx & = frac {N} {M ^ 2} int _ { mathbb {R} ^ {n-1}} int _ {- 1} ^ {1} zeta (x ‘, frac {x_n} {M}) e ^ {- pN frac {x_n} {M ^ 2}} dx_n dx ‘ aim {align *}

I attempted utilizing the combination by components however I could not get the outcome I wanted.

Any ameliorate or counsel is appreciated. Thank you very mighty.

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