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Nested Function


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 Nest[f, expr, n], and a listing of incremental nestings is implemented as NestList[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

inte^xdx=e^x
(1)
inte^(e^x)dx=Ei(e^x),
(2)

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

intlnxdx=x(lnx-1)
(3)
intlnlnxdx=xlnlnx-li(x),
(4)

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 Web Resource. https://mathworld.wolfram.com/NestedFunction.html

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