In mathematics, the Legendre chi function is a special function whose Taylor series is also a Dirichlet series, given by χ ν ( z ) = ∑ k = 0 ∞ z 2 k +...

_{n=1}^{\infty }{\frac {1}{n^{3}}}x^{n}\end{aligned}}} The Legendre chi functions are defined as follows: χ 2 ( x ) = ∑ n = 0 ∞ 1 ( 2 n + 1 ) 2 x...

Dirichlet beta function Dirichlet L-function Hurwitz zeta function Legendre chi function Lerch transcendent Polylogarithm and related functions: Incomplete...

the Legendre chi function χ ν {\displaystyle \chi _{\nu }} as C ν ( x ) = Re χ ν ( e i x ) {\displaystyle C_{\nu }(x)=\operatorname {Re} \,\chi _{\nu...

other series for the zeta-function-related cases of the Legendre chi function, the polygamma function, and the Riemann zeta function include χ 1 ( z ) = ∑...

{\operatorname {Ti} _{n}(t)}{t}}dt,} which explains the function name. The Legendre chi function χs(z) (Lewin 1958, Ch. VII § 1.1; Boersma & Dempsey 1992)...

Gauss–Legendre algorithm Gauss–Legendre method Gauss–Legendre quadrature Legendre (crater) Legendre chi function Legendre duplication formula Legendre–Papoulis...

Dirichlet beta function. The inverse tangent integral is related to the Legendre chi function χ 2 ( x ) = x + x 3 3 2 + x 5 5 2 + ⋯ {\textstyle \chi _{2}(x)=x+{\frac...

distribution with three degrees of freedom). The probability density function (pdf) of the chi-distribution is f ( x ; k ) = { x k − 1 e − x 2 / 2 2 k / 2 −...

1 2 ( χ ) {\displaystyle Q_{m-{\frac {1}{2}}}(\chi )} is the odd-half-integer degree Legendre function of the second kind, which is a toroidal harmonic...

}(x)&=S_{\nu }(1-x).\end{aligned}}} They are related to the Legendre chi function χ ν {\displaystyle \chi _{\nu }} as C ν ( x ) = Re ⁡ χ ν ( e i x ) S ν ( x )...

Stickelberger's theorem. When χ is the Legendre symbol, J ( χ , χ ) = − χ ( − 1 ) = ( − 1 ) p + 1 2 . {\displaystyle J(\chi ,\chi )=-\chi (-1)=(-1)^{\frac {p+1}{2}}\...

{\tfrac {1}{2}})} The Legendre chi function: χ s ( z ) = ∑ k = 0 ∞ z 2 k + 1 ( 2 k + 1 ) s = z 2 s Φ ( z 2 , s , 1 2 ) {\displaystyle \chi _{s}(z)=\sum _{k=0}^{\infty...

related branches of mathematics, a complex-valued arithmetic function χ : Z → C {\displaystyle \chi :\mathbb {Z} \rightarrow \mathbb {C} } is a Dirichlet character...

time. The method of least squares was published by Legendre in 1805, and by Gauss in 1809. Legendre and Gauss both applied the method to the problem of...

product of the Riemann zeta function and a certain Dirichlet L-function The Jacobi symbol is a generalization of the Legendre symbol; the main difference...

theory, an arithmetic, arithmetical, or number-theoretic function is generally any function whose domain is the set of positive integers and whose range...

{\displaystyle G(\chi )=\mu \left({\frac {N}{N_{0}}}\right)\chi _{0}\left({\frac {N}{N_{0}}}\right)G\left(\chi _{0}\right)} where μ is the Möbius function. Consequently...

parameters and the observed data. The method was first proposed by Adrien-Marie Legendre in 1805 and further developed by Carl Friedrich Gauss. The method of least...

convex function. Some operations in convex analysis, such as the Legendre transform automatically produce closed convex functions. The Legendre transform...

index equal to zero are proportional to spherical harmonics and associated Legendre polynomials, normalized to unity and with Condon and Shortley phase convention:...

Vooren. Boersma, J.; Dempsey, J.P. (1992). "On the evaluation of Legendre's chi-function" (PDF). Mathematics of Computation. 59 (199): 157–163. doi:10.2307/2152987...

( a ) ζ a {\displaystyle \sum \chi (a)\zeta ^{a}} where χ ( a ) {\displaystyle \chi (a)} here stands for the Legendre symbol (a/p), and the sum is taken...

extension of the domain is necessary for defining L functions. See Legendre symbol#Properties of the Legendre symbol for examples Lemmermeyer, pp 111–end Davenport...

F(n)=n(n+1)} and χ a Legendre symbol. Here the sum can be evaluated (as −1), a result that is connected to the local zeta-function of a conic section....

Practice. Boca Raton: CRC Press. p. 204. ISBN 9781584886167. Legendre, Pierre; Legendre, Louis (2012). Numerical Ecology. Amsterdam: Elsevier. p. 465...

continued-fraction representation of the tangent function. French mathematician Adrien-Marie Legendre proved in 1794 that π2 is also irrational. In 1882...

\zeta _{p}=\exp(2\pi i/p)} . Equivalently, we can write this using the Legendre symbol as g ( a ; p ) = ∑ n = 0 p − 1 ( 1 + ( n p ) ) ζ p a n . {\displaystyle...

dynamics and cost functions DNSS point — initial state for certain optimal control problems with multiple optimal solutions Legendre–Clebsch condition...

corresponding to the function of bounded variation χ [ a , b ] ( x ) f ( x ) {\displaystyle \chi _{[a,b]}(x)f(x)} , and functions f ~ , φ ~ {\displaystyle...