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Integral cosine

Integral cosine plot for 0 < x ≤ 8π.

The integral cosine is a special function defined by the integral ^[1]

\operatorname {Ci} (x)=-\int \limits _{x}^{\infty }{\frac {\cos t}{t}}dt

{\ displaystyle \ operatorname {Ci} (x) = - \ int \ limits _ {x} ^ {\ infty} {\ frac {\ cos t} {t}} dt}

{\ displaystyle \ operatorname {Ci} (x) = - \ int \ limits _ {x} ^ {\ infty} {\ frac {\ cos t} {t}} dt}

or:

\gamma +\ln x+\int \limits _{0}^{x}{\frac {\cos t-1}{t}}\,dt

{\ displaystyle \ gamma + \ ln x + \ int \ limits _ {0} ^ {x} {\ frac {\ cos t-1} {t}} \, dt}

{\ displaystyle \ gamma + \ ln x + \ int \ limits _ {0} ^ {x} {\ frac {\ cos t-1} {t}} \, dt}

Where $\gamma$ ${\ displaystyle \ gamma}$ $\ gamma$ - Euler-Mascheroni constant .

Other definitions are sometimes used:

\operatorname {Cin} (x)=\int \limits _{0}^{x}{\frac {1-\cos t}{t}}\,dt

{\ displaystyle \ operatorname {Cin} (x) = \ int \ limits _ {0} ^ {x} {\ frac {1- \ cos t} {t}} \, dt}

{\ displaystyle \ operatorname {Cin} (x) = \ int \ limits _ {0} ^ {x} {\ frac {1- \ cos t} {t}} \, dt}

\operatorname {Cin} (x)=\gamma +\ln x-\operatorname {Ci} (x).

{\ displaystyle \ operatorname {Cin} (x) = \ gamma + \ ln x- \ operatorname {Ci} (x).}

{\ displaystyle \ operatorname {Cin} (x) = \ gamma + \ ln x- \ operatorname {Ci} (x).}

It is also possible to determine the integral cosine through the integral exponential function by analogy with the usual cosine ^[2] :

\operatorname {Ci} (x)={\frac {1}{2}}\left(\operatorname {Ei} (ix)+\operatorname {Ei} (-ix)\right)

{\ displaystyle \ operatorname {Ci} (x) = {\ frac {1} {2}} \ left (\ operatorname {Ei} (ix) + \ operatorname {Ei} (-ix) \ right)}

{\ displaystyle \ operatorname {Ci} (x) = {\ frac {1} {2}} \ left (\ operatorname {Ei} (ix) + \ operatorname {Ei} (-ix) \ right)}

The integral cosine was introduced by Lorenzo Mascheroni in 1790 .

Content

Properties

The integral cosine can be represented as a series:

\operatorname {Ci} (x)=\gamma +\ln x-{\frac {x^{2}}{2\cdot 2!}}+{\frac {x^{4}}{4\cdot 4!}}-{\frac {x^{6}}{6\cdot 6!}}+\cdots =\gamma +\ln x+\sum _{n=1}^{\infty }{\frac {(-1)^{n}x^{2n}}{(2n)!(2n)}}

{\ displaystyle \ operatorname {Ci} (x) = \ gamma + \ ln x - {\ frac {x ^ {2}} {2 \ cdot 2!}} + {\ frac {x ^ {4}} {4 \ cdot 4!}} - {\ frac {x ^ {6}} {6 \ cdot 6!}} + \ cdots = \ gamma + \ ln x + \ sum _ {n = 1} ^ {\ infty} {\ frac {(-1) ^ {n} x ^ {2n}} {(2n)! (2n)}}}

See also

Integral sine
Integral exponential function
Integral Logarithm

Notes

↑ Korn G., Korn T. Handbook of mathematics for scientists and engineers. // M .: Nauka, 1968 .-- p. 625
↑ Beitman G., Erdeyi A. Higher transcendental functions, vol. 2 // M .: Nauka, 1974. - p. 149

Literature

Mathematical Encyclopedic Dictionary, M. 1995, p. 238
Weisstein, Eric W. Cosine Integral on the Wolfram MathWorld website.

Source - https://ru.wikipedia.org/w/index.php?title=Cosine Integral&oldid = 82917086

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