A multidimensional Jordan residue lemma with an application to Mellin-Barnes integrals

The classical Jordan lemma states that if a function ψ is continuous on the real axis with a holomorphic continuation to the upper half plane Π+ = { z = x + iy;y > 0} except for a finite number of points {a} ⊂ Π+, and if ψ(z) tends to zero as |z| → ∞ in the closed half plane Π+, then 0.1 % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qmaeaacq % aHipqEcaGGOaGaamiEaiaacMcacaWGLbWaaWbaaSqabeaacaWGPbGa % amyyaiaadIhaaaGccaWGKbGaamiEaiabg2da9iaaikdacqaHapaCca % WGPbWaaabuaeaacaWGYbGaamyzaiaadohadaWgaaWcbaGaamyyaaqa % baGccaaMe8UaeqyYdCNaaiilaaWcbaWaaiWaaeaacaWGHbaacaGL7b % GaayzFaaaabeqdcqGHris5aaWcbaGaeyOeI0IaeyOhIukabaGaeyOh % IukaniabgUIiYdaaaa!578E! $$ \int_{ - \infty }^\infty {\psi (x){e^{iax}}dx = 2\pi i\sum\limits_{\left\{ a \right\}} {re{s_a}\;\omega ,} } $$ where α is an arbitrary positive number and ω denotes the differential form ψ(z)e iaz dz.

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