In complex analysis, the Phragmén–Lindelöf principle (or method), first formulated by Lars Edvard Phragmén (1863–1937) and Ernst Leonard Lindelöf (1870–1946) in 1908, is a technique which employs an auxiliary, parameterized function to prove the boundedness of a holomorphic function (i.e, ) on an unbounded domain when an additional (usually mild) condition constraining the growth of on is given. It is a generalization of the maximum modulus principle, which is only applicable to bounded domains. Background In the theory of complex functions, it is known that the modulus (absolute value) of a holomorphic (complex differentiable) function in the interior of a bounded region is bounded by its modulus on the boundary of the region. More precisely, if a non-constant function is holomorphic in a bounded region and continuous on its closure , then for all . This is known as the maximum modulus principle. (In fact, since is compact and is continuous, there actually exists some such that .) The maximum modulus principle is generally used to conclude that a holomorphic function is bounded in a region after showing that it is bounded on its boundary. However, the maximum modulus principle cannot be applied to an unbounded region of the complex plane. As a concrete example, let us examine the behavior of the holomorphic function in the unbounded strip . Although , so that is bounded on boundary , grows rapidly without bound when along the positive real axis. The difficulty here stems from the extremely fast growth of along the positive real axis. If the growth rate of is guaranteed to not be "too fast," as specified by an appropriate growth condition, the Phragmén–Lindelöf principle can be applied to show that boundedness of on the region's boundary implies that is in fact bounded in the whole region, effectively extending the maximum modulus principle to unbounded regions. Outline of the technique Suppose we are given a holomorphic function and an unbounded region , and we want to show that on . In a typical Phragmén–Lindelöf argument, we introduce a certain multiplicative factor satisfying to "subdue" the growth of . In particular, is chosen such that (i): is holomorphic for all and on the boundary of an appropriate bounded subregion ; and (ii): the asymptotic behavior of allows us to establish that for (i.e., the unbounded part of outside the closure of the bounded subregion). This allows us to apply the maximum