The most common form of cosine integral is
(1)
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(2)
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(3)
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(4)
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where is the exponential integral, is the En-function, and is the Euler-Mascheroni constant.
is returned by the Wolfram Language command CosIntegral[x], and is also commonly denoted .
has the series expansion
(5)
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(Havil 2003, p. 106; after inserting a minus sign in the definition).
The derivative is
(6)
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and the integral is
(7)
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has zeros at 0.616505, 3.38418, 6.42705, .... Extrema occur when
(8)
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or , or , , , ..., which are alternately maxima and minima. At these points, equals 0.472001, , 0.123772, .... Inflection points occur when
(9)
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which simplifies to
(10)
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which has solutions 2.79839, 6.12125, 9.31787, ....
The related function
(11)
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(12)
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is sometimes also defined.
To find a closed form for an integral power of a cosine function, use integration by parts to obtain
(13)
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(14)
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Now, if is even so , then
(15)
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(16)
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On the other hand, if is odd so , then
(17)
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Now let ,
(18)
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The general result is then
(19)
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The infinite integral of a cosine times a Gaussian can also be done in closed form,
(20)
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