Nested Function

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A composition of a function 👁 f degreesf
with itself gives a nested function 👁 f(f(x))
, 👁 f degreesf degreesf
which gives 👁 f(f(f(x))
, etc. Function nesting is implemented in the Wolfram Language as [f, expr, n], and a listing of incremental nestings is implemented as [f, expr, n].

Integrals of nested functions rapidly become difficult to find in exact form. For example, the first two nestings of the exponential function gives

where 👁 Ei(x)
is the exponential integral, the first two nestings of the natural logarithm give

where 👁 li(x)
is the logarithmic integral.

See also

Composition, Integral, Iterated Map, Iteration, Nested Logarithm

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Cite this as:

Weisstein, Eric W. "Nested Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/NestedFunction.html

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