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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, the Phragmén Lindelöf principle is a 1908 extension by Lars Edvard Phragmén (1863 1937) and Ernst Leonard Lindelöf of the maximum modulus principle of complex analysis, to unbounded domains. In complex function theory it is known that if a function f is holomorphic in a bounded domain D, and is continuous on the boundary of D, then the maximum of f must be attained on the boundary of D. If, however, the region D is not bounded, then this is no longer true, as may be seen by examining the function g(z) = exp(exp(z)) in the strip / 2 Im{z} / 2. The difficulty here is that the function g tends to infinity ''very'' rapidly as z tends to infinity along the positive real axis.