The natural boundary of analyticity of the function with respect to is the set boundary.
Gives a list of lists of the branch points (if present, including infinity) of the function f over the complex ‐plane.
The ramification index for function in the branch point .
The set of poles (with their orders) and essential singularities of with respect to . (The order of essential singularity is ∞.)
The list of the (parametrized) intervals where the function is discontinuous over the complex ‐plane.
Asymptotic relation that reflects the boundedness of near point .
Asymptotic relation that reflects the boundedness of near point .
The asymptotic expansion of the function at the point that includes terms of the asymptotic expansions of the regular and exponential components: . .
Sum of terms excluding the term .
Limit of the finite sum (infinite sum): .
Limit of the finite sum (infinite sum): .
Sum over all solutions of the equation .
Sum over the set .
Sum of over all divisors of n.
Multiple sum of function over the sets .
Limit of finite product .
Product over set .
Product of over all divisors of n.
Indefinite integral (antiderivative) of function . Inverse operation to differentiation: .
Multiple definite integral of the function by the intervals .
Cauchy principal value of a singular integral: Cauchy principal value of a singular integral:
The special contour , which is used in the definition of the Meijer G function and its numerous particular cases. There are three possibilities for the contour : (i) runs from γⅈ ∞ to γ+ⅈ ∞ (where ) so that all poles of , are to the left of ℒ, and all poles of , , are to the right.
This contour can be a straight line if (then ). (In this case, the integral converges if , . If , then must be real and positive, and the additional condition should be added. (ii) is a loop on the left side of the complex plane, starting and ending at ∞ and encircling all poles of ,, once in the clockwise direction, but none of the poles of , .
(In this case, the integral converges if and either , or and , or and and both and .) (iii) is a loop on the right side of the complex plane, starting and ending at +∞ and encircling all poles of , , once in the counterclockwise direction, but none of the poles of , .
(In this case, the integral converges if and either , or and , or and and both and .)
Wigner integral transform: . (If this integral does not converge, the value of is defined in the sense of generalized functions.)
A finite continued fraction
Limit of the finite continued fraction .
The matrix with elements .
The determinant of the matrix with elements .
