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Mapping theorems

The question often comes up, of designing a function taking given values at specified places. Naturally the answer depends on knowing what kinds of values, and at what places. For example, assigning values which violated the maximum modulus principle or Schwartz's lemma (without introducing singularities) would be futile.

Three levels at which the task may be considered are: i) mapping points, ii) mapping lines, and iii) mapping regions. The first is solved by interpolation, the second by the Schwartz-Christoffel process, and the third by the Riemann mapping theorem.