This is a list of indefinite sums (also known as antidifferences) of various functions. An indefinite sum ∑ x f ( x ) {\textstyle \sum _{x}f(x)} is the inverse of the forward difference operator Δ {\displaystyle \Delta } , defined as Δ f ( x ) = f ( x + 1 ) − f ( x ) {\displaystyle \Delta f(x)=f(x+1)-f(x)} . It satisfies the relation
Δ ∑ x f ( x ) = f ( x ) . {\displaystyle \Delta \sum _{x}f(x)=f(x).}
The operator is defined only up to an additive periodic function with period 1.
Antidifferences of rational functions For positive integer exponents, Faulhaber's formula can be used. Note that x {\displaystyle x} in the result of Faulhaber's formula must be replaced with x − 1 {\displaystyle x-1} due to the offset, as Faulhaber's formula finds ∇ − 1 {\displaystyle \nabla ^{-1}} rather than Δ − 1 {\displaystyle \Delta ^{-1}} . For negative integer exponents, the indefinite sum is closely related to the polygamma function:
∑ x 1 x a = ( − 1 ) a − 1 ψ ( a − 1 ) ( x ) ( a − 1 ) ! + C , a ∈ N {\displaystyle \sum _{x}{\frac {1}{x^{a}}}={\frac {(-1)^{a-1}\psi ^{(a-1)}(x)}{(a-1)!}}+C,\,a\in \mathbb {N} }
For fractions not listed in this section, one may use the polygamma function with partial fraction decomposition. More generally,
∑ x x a = { B a + 1 ( x ) a + 1 + C , if a ≠ − 1 ψ ( x ) + C , if a = − 1 = { − ζ ( − a , x ) + C , if a ≠ − 1 ψ ( x ) + C , if a = − 1 {\displaystyle \sum _{x}x^{a}={\begin{cases}{\frac {B_{a+1}(x)}{a+1}}+C,&{\text{if }}a\neq -1\\\psi (x)+C,&{\text{if }}a=-1\end{cases}}={\begin{cases}-\zeta (-a,x)+C,&{\text{if }}a\neq -1\\\psi (x)+C,&{\text{if }}a=-1\end{cases}}}
where B a ( x ) {\displaystyle B_{a}(x)} are the Bernoulli polynomials, ζ ( s , a ) {\displaystyle \zeta (s,a)} is the Hurwitz zeta function, and ψ ( z ) {\displaystyle \psi (z)} is the digamma function. This is related to the generalized harmonic numbers. As the generalized harmonic numbers use reciprocal powers, a {\displaystyle a} must be substituted for − a {\displaystyle -a} , and the most common form uses the inverse of the backward difference offset:
∇ − 1 x a = H x ( − a ) = ζ ( − a ) − ζ ( − a , x + 1 ) . {\displaystyle \nabla ^{-1}x^{a}={H_{x}^{(-a)}}=\zeta (-a)-\zeta (-a,x+1).}
Here, ζ ( − a ) {\displaystyle \zeta (-a)} is the constant C {\displaystyle C} . The Bernoulli polynomials are also related via a partial derivative with respect to x {\displaystyle x} :
∂ ∂ x ( ∑ x x a ) = B a ( x ) = − a ζ ( 1 − a , x ) . {\displaystyle {\frac {\partial }{\partial x}}\left(\sum _{x}x^{a}\right)=B_{a}(x)=-a\zeta (1-a,x).}
This relationship can be expressed via the inverse backward difference operator as:
∂ ∂ x ( ∇ − 1 x a ) | x = 0 = − a ζ ( 1 − a , x + 1 ) | x = 0 = − a ζ ( 1 − a ) = B a . {\displaystyle {\frac {\partial }{\partial x}}\left(\nabla ^{-1}x^{a}\right){\bigg |}_{x=0}=-a\zeta (1-a,x+1){\bigg |}_{x=0}=-a\zeta (1-a)=B_{a}.}
Further generalization comes from use of the Lerch transcendent:
∑ x z x ( x + a ) s = − z x Φ ( z , s , x + a ) + C , {\displaystyle \sum _{x}{\frac {z^{x}}{(x+a)^{s}}}=-z^{x}\,\Phi (z,s,x+a)+C,}
which generalizes the generalized harmonic numbers as z Φ ( z , s , a + 1 ) − z x + 1 Φ ( z , s , x + 1 + a ) {\displaystyle z\Phi \left(z,s,a+1\right)-z^{x+1}\Phi \left(z,s,x+1+a\right)} when taking ∇ − 1 {\displaystyle \nabla ^{-1}} .
∑ x B a ( x ) = ( x − 1 ) B a ( x ) − a a + 1 B a + 1 ( x ) + C {\displaystyle \sum _{x}B_{a}(x)=(x-1)B_{a}(x)-{\frac {a}{a+1}}B_{a+1}(x)+C}
Antidifferences of exponential functions
∑ x a x = a x a − 1 + C {\displaystyle \sum _{x}a^{x}={\frac {a^{x}}{a-1}}+C}
Antidifferences of logarithmic functions
∑ x log b x = log b Γ ( x ) + C {\displaystyle \sum _{x}\log _{b}x=\log _{b}\Gamma (x)+C}
∑ x log b a x = log b ( a x − 1 Γ ( x ) ) + C {\displaystyle \sum _{x}\log _{b}ax=\log _{b}(a^{x-1}\Gamma (x))+C}
Antidifferences of hyperbolic functions
∑ x sinh a x = 1 2 csch ( a 2 ) cosh ( a 2 − a x ) + C {\displaystyle \sum _{x}\sinh ax={\frac {1}{2}}\operatorname {csch} \left({\frac {a}{2}}\right)\cosh \left({\frac {a}{2}}-ax\right)+C}
∑ x cosh a x = 1 2 csch ( a 2 ) sinh ( a x − a 2 ) + C {\displaystyle \sum _{x}\cosh ax={\frac {1}{2}}\operatorname {csch} \left({\frac {a}{2}}\right)\sinh \left(ax-{\frac {a}{2}}\right)+C}
∑ x tanh a x = 1 a ψ e a ( x − i π 2 a ) + 1 a ψ e a ( x + i π 2 a ) − x + C {\displaystyle \sum _{x}\tanh ax={\frac {1}{a}}\psi _{e^{a}}\left(x-{\frac {i\pi }{2a}}\right)+{\frac {1}{a}}\psi _{e^{a}}\left(x+{\frac {i\pi }{2a}}\right)-x+C}
where ψ q ( x ) {\displaystyle \psi _{q}(x)} is the q-digamma function.
Antidifferences of trigonometric functions
∑ x sin a x = − 1 2 csc ( a 2 ) cos ( a 2 − a x ) + C , a ≠ 2 n π {\displaystyle \sum _{x}\sin ax=-{\frac {1}{2}}\csc \left({\frac {a}{2}}\right)\cos \left({\frac {a}{2}}-ax\right)+C\,,\,\,a\neq 2n\pi }
∑ x cos a x = 1 2 csc ( a 2 ) sin ( a x − a 2 ) + C , a ≠ 2 n π {\displaystyle \sum _{x}\cos ax={\frac {1}{2}}\csc \left({\frac {a}{2}}\right)\sin \left(ax-{\frac {a}{2}}\right)+C\,,\,\,a\neq 2n\pi }
∑ x sin 2 a x = x 2 + 1 4 csc ( a ) sin ( a − 2 a x ) + C , a ≠ n π {\displaystyle \sum _{x}\sin ^{2}ax={\frac {x}{2}}+{\frac {1}{4}}\csc(a)\sin(a-2ax)+C\,\,,\,\,a\neq n\pi }
∑ x cos 2 a x = x 2 − 1 4 csc ( a ) sin ( a − 2 a x ) + C , a ≠ n π {\displaystyle \sum _{x}\cos ^{2}ax={\frac {x}{2}}-{\frac {1}{4}}\csc(a)\sin(a-2ax)+C\,\,,\,\,a\neq n\pi }
∑ x tan a x = i x − 1 a ψ e 2 i a ( x − π 2 a ) + C , a ≠ n π 2 {\displaystyle \sum _{x}\tan ax=ix-{\frac {1}{a}}\psi _{e^{2ia}}\left(x-{\frac {\pi }{2a}}\right)+C\,,\,\,a\neq {\frac {n\pi }{2}}}
where ψ q ( x ) {\displaystyle \psi _{q}(x)} is the q-digamma function.
∑ x tan x = i x − ψ e 2 i ( x + π 2 ) + C = − ∑ k = 1 ∞ ( ψ ( k π − π 2 + 1 − x ) + ψ ( k π − π 2 + x ) − ψ ( k π − π 2 + 1 ) − ψ ( k π − π 2 ) ) + C {\displaystyle {\begin{aligned}\sum _{x}\tan x&=ix-\psi _{e^{2i}}\left(x+{\frac {\pi }{2}}\right)+C\\&=-\sum _{k=1}^{\infty }\left(\psi \left(k\pi -{\frac {\pi }{2}}+1-x\right)+\psi \left(k\pi -{\frac {\pi }{2}}+x\right)\right.\\&\quad \left.-\psi \left(k\pi -{\frac {\pi }{2}}+1\right)-\psi \left(k\pi -{\frac {\pi }{2}}\right)\right)+C\end{aligned}}}
∑ x cot a x = − i x − i ψ e 2 i a ( x ) a + C , a ≠ n π 2 {\displaystyle \sum _{x}\cot ax=-ix-{\frac {i\psi _{e^{2ia}}(x)}{a}}+C\,,\,\,a\neq {\frac {n\pi }{2}}}
The antidifference of the normalized sinc function can be obtained by applying the Abel–Plana formula presented in Candelpergher with the shift x ↦ x − 1 {\displaystyle x\mapsto x-1} , the condition F ( 0 ) = 0 {\displaystyle F(0)=0} , and recurrence of F ( x + 1 ) − F ( x ) = f ( x ) {\displaystyle F(x+1)-F(x)=f(x)} . Using the reflection formula for the digamma function, this simplifies to:
∑ x sinc x = sinc ( x − 1 ) ( 1 2 + ( x − 1 ) × ( ln ( 2 ) + ψ ( x − 1 2 ) + ψ ( 1 − x 2 ) 2 − ψ ( x − 1 ) + ψ ( 1 − x ) 2 ) ) + 1 2 + C {\displaystyle {\begin{aligned}\sum _{x}\operatorname {sinc} x&=\operatorname {sinc} (x-1)\left({\frac {1}{2}}+(x-1)\right.\\&\quad \left.\times \left(\ln(2)+{\frac {\psi ({\frac {x-1}{2}})+\psi ({\frac {1-x}{2}})}{2}}\right.\right.\\&\quad \quad \left.\left.-{\frac {\psi (x-1)+\psi (1-x)}{2}}\right)\right)+{\frac {1}{2}}+C\end{aligned}}}
Period rules If T {\displaystyle T} is a period of function f ( x ) {\displaystyle f(x)} then
∑ x f ( T x ) = x f ( T x ) + C . {\displaystyle \sum _{x}f(Tx)=xf(Tx)+C.}
If T {\displaystyle T} is an antiperiod of function f ( x ) {\displaystyle f(x)} , that is f ( x + T ) = − f ( x ) {\displaystyle f(x+T)=-f(x)} then
∑ x f ( T x ) = − 1 2 f ( T x ) + C . {\displaystyle \sum _{x}f(Tx)=-{\frac {1}{2}}f(Tx)+C.}
Antidifferences of special functions
∑ x ψ ( x ) = ( x − 1 ) ψ ( x ) − x + C {\displaystyle \sum _{x}\psi (x)=(x-1)\psi (x)-x+C}
∑ x Γ ( x ) = ( − 1 ) x + 1 Γ ( x ) Γ ( 1 − x , − 1 ) e + C {\displaystyle \sum _{x}\Gamma (x)=(-1)^{x+1}\Gamma (x){\frac {\Gamma (1-x,-1)}{e}}+C}
where Γ ( s , x ) {\displaystyle \Gamma (s,x)} is the incomplete gamma function.
∑ x ( x ) a = ( x ) a + 1 a + 1 + C {\displaystyle \sum _{x}(x)_{a}={\frac {(x)_{a+1}}{a+1}}+C}
where ( x ) a {\displaystyle (x)_{a}} is the falling factorial.
∑ x sexp a ( x ) = ln a ( sexp a ( x ) ) ′ ( ln a ) x + C {\displaystyle \sum _{x}\operatorname {sexp} _{a}(x)=\ln _{a}{\frac {(\operatorname {sexp} _{a}(x))'}{(\ln a)^{x}}}+C}
(see super-exponential function)
References