The Collaborative International Dictionary of English v.0.48:

Exigent \Ex`i*gent\, a. [L. exigens, -entis, p. pr. of exigere
   to drive out or forth, require, exact. See Exact.]
   Exacting or requiring immediate aid or action; pressing;
   critical. "At this exigent moment." --Burke.
   [1913 Webster]

The Collaborative International Dictionary of English v.0.48:

Exigent \Ex"i*gent\, n.
   1. Exigency; pressing necessity; decisive moment. [Obs.]
      [1913 Webster]

            Why do you cross me in this exigent?  --Shak.
      [1913 Webster]

   2. (o. Eng. Law) The name of a writ in proceedings before
      outlawry. --Abbott.
      [1913 Webster]

WordNet (r) 3.0 (2006):

exigent
    adj 1: demanding attention; "clamant needs"; "a crying need";
           "regarded literary questions as exigent and momentous"-
           H.L.Mencken; "insistent hunger"; "an instant need" [syn:
           clamant, crying, exigent, insistent, instant]
    2: requiring precise accuracy; "an exacting job"; "became more
       exigent over his pronunciation" [syn: exigent, exacting]

Moby Thesaurus II by Grady Ward, 1.0:

95 Moby Thesaurus words for "exigent":
   Spartan, Spartanic, accurate, acute, astringent, attentive,
   austere, authoritarian, burdensome, burning, charged, choiceless,
   clamant, clamorous, climacteric, close, compelling, compulsory,
   conscientious, correct, critical, crucial, crying, decisive,
   delicate, demanding, detailed, dour, draining, emergent, exact,
   exacting, exorbitant, exquisite, extortionate, fine, finical,
   finicking, finicky, fussy, grasping, grievous, grim, harsh,
   high-pressure, high-priority, imperative, imperious, importunate,
   insistent, instant, kairotic, loaded, loud, mandatory, menacing,
   meticulous, minute, narrow, necessary, necessitous, nice,
   obligatory, oppressive, particular, persistent, pertinacious,
   pivotal, precise, precisianistic, precisionistic, pregnant,
   pressing, punctilious, punctual, refined, religious, rigid,
   rigorous, rugged, scrupulous, scrutinizing, severe, stern, strict,
   stringent, subtle, superincumbent, taxing, threatening, tough,
   unsparing, urgent, weighty, without choice



Bouvier's Law Dictionary, Revised 6th Ed (1856):

EXIGENT, or EXIGI FACIAS, practice. A writ issued in the course of
proceedings to outlawry, deriving its name and application from the
mandatory words found therein, signifying, "that you cause to be exacted or
required; and it is that proceeding in an outlawry which, with the writ of
proclamation, issued at the same time, immediately precedes the writ of
capias utlagatum. 2 Virg. Cas. 244.