The exponential integral is a special function defined on the complex
plane. For real, nonzero values of x, the exponential integral Ei(x)
can be defined as Ei(x) = int(from -∞ to x) e^t/t dx. The function
is given as a special function because int e^t/t dx is not an
elementary function. The definition above can be used for positive
values of x, but the integral has to be understood in terms of the
Cauchy principal value, due to the singularity in the integrand at
zero. For complex values of the argument, the definition becomes
ambiguous due to branch points at 0 and ∞. In general, a branch cut
is taken on the negative real axis and Ei can be defined by analytic
continuation elsewhere on the complex plane.
Arieta Tekaura
20.08.11, 06:38
The answer is good, except for the last part. What is Ei(x)? How to
work it out? Is it a constant or something else? Can you explain what
is the exponential integral?
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The exponential integral is a special function defined on the complex plane. For real, nonzero values of x, the exponential integral Ei(x) can be defined as Ei(x) = int(from -∞ to x) e^t/t dx. The function is given as a special function because int e^t/t dx is not an elementary function. The definition above can be used for positive values of x, but the integral has to be understood in terms of the Cauchy principal value, due to the singularity in the integrand at zero. For complex values of the argument, the definition becomes ambiguous due to branch points at 0 and ∞. In general, a branch cut is taken on the negative real axis and Ei can be defined by analytic continuation elsewhere on the complex plane.
The answer is good, except for the last part. What is Ei(x)? How to work it out? Is it a constant or something else? Can you explain what is the exponential integral?
Leave a comment