Integral cosine

Course of the integral cosine (in green) in the range 0 ≤ x ≤ 8π (lower curve).
For comparison in blue the integral sine (upper curve)

The integral cosine is a function in whose function specification an integral and the cosine function appear. This integral function cannot be represented by elementary methods without an integral.

The integral cosine is defined as:

${\displaystyle {\rm {Ci}}(x):=\gamma +\ln x+\int _{0}^{x}{\frac {\cos t-1}{t}}\,dt\quad =-\int _{x}^{\infty }{\frac {\cos t}{t}}\,dt}$

Where is the Euler-Mascheroni constant ${\displaystyle \gamma =0{,}577215...}$

properties

• The integral occurring in the definition is also referred to as: ${\displaystyle \operatorname {Cin} }$

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

with the relationship:

${\displaystyle {\rm {Cin}}(x)=\gamma +\ln x-{\rm {Ci}}(x)\,}$

• Analogous to the derivative of the integral sine Si (x):

${\displaystyle \mathrm {Si} '(x)={\frac {\sin x}{x}}}$

applies:

${\displaystyle \mathrm {Ci} '(x)={\frac {\cos(x)}{x}}}$

• Analogous to the complex Euler formula definition of the cosine

${\displaystyle \cos x={\frac {1}{2}}\left(\mathrm {e} ^{\mathrm {i} x}+\mathrm {e} ^{-\mathrm {i} x}\right)}$

holds with the integral exponential function ${\displaystyle \mathrm {Ei} }$

${\displaystyle \mathrm {Ci} (x)={\frac {1}{2}}\left(\mathrm {Ei} (\mathrm {i} \ x)+\mathrm {Ei} (-\mathrm {i} \ x)\right)}$.

• A series that converges everywhere can be given:

${\displaystyle \mathrm {Ci} \left(x\right)=\gamma +\ln x-{\frac {x^{2}}{2!\cdot 2}}+{\frac {x^{4}}{4!\cdot 4}}-\cdots \quad =\gamma +\ln x+\sum _{k=1}^{\infty }{\frac {(-1)^{k}x^{2k}}{(2k)!\cdot 2k}}}$.

Note: In various formulas, the integral cosine is defined with the opposite sign .

Closely related is the integral sine , which together with the integral cosine forms a clothoid in parametric representation . ${\displaystyle Si(x)}$ ${\displaystyle Ci(x)}$

See also

• Integral exponential function
• Integral sine

Web links

• Eric W. Weisstein : Cosine Integral . In: MathWorld (English).