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Lacunary value

Last updated August 30, 2025

In complex analysis, a subfield of mathematics, a lacunary value or gap of a complex-valued function defined on a subset of the complex plane is a complex number which is not in the image of the function.^[1]

More specifically, given a subset X of the complex plane C and a function f : X → C, a complex number z is called a lacunary value of f if z ∉ image(f).

Note, for example, that 0 is the only lacunary value of the complex exponential function. The two Picard theorems limit the number of possible lacunary values of certain types of holomorphic functions.

References

↑ Clark, Douglas N., ed. (1999). Dictionary of Analysis, Calculus, and Differential Equations. Comprehensive dictionary of mathematics. Vol. 1. CRC Press. pp. 97–98. ISBN 9780849303203..

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[1] Clark, Douglas N., ed. (1999). Dictionary of Analysis, Calculus, and Differential Equations. Comprehensive dictionary of mathematics. Vol. 1. CRC Press. pp. 97–98. ISBN 9780849303203..

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