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    $\begingroup$ isn't this also the basic idea behind zeta function regularization (replacing a divergent series by the finite value obtained from analytically continuing the series from a convergent parameter domain)? $\endgroup$ Commented Dec 5, 2013 at 9:20
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    $\begingroup$ Also, one example is to do path integrals in imaginary time, then aggressively exploit the the fact that physical quantities are holomorphic functions of momenta to "Wick rotate" the results back to real time. $\endgroup$ Commented Dec 5, 2013 at 9:24
  • $\begingroup$ Related: physics.stackexchange.com/search?q=is%3Aq+analytic+continuation $\endgroup$ Commented Dec 5, 2013 at 10:41
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    $\begingroup$ Interestingly, "most" holomorphic functions cannot be continued beyond their specified domain - they have natural boundaries there. See, for example, this MathOverflow thread. $\endgroup$ Commented Feb 28, 2014 at 10:05