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Notations

Listing of the Mathematical Notations used in the Mathematical Functions Website












Notations





Operations


The function is defined on domain , and it acts from this domain to domain .

The natural boundary of analyticity of the function with respect to is the set boundary.

represents the branch cuts branchCuts of the function with respect to . Each branch cut is of the form {interval, direction} indicating a branch cut along interval and continuity of from the direction direction.

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.

is the main term of asymptotic expansion of at that reflects the property: .

Asymptotic relation that reflects the boundedness of near point .

Asymptotic relation that reflects the boundedness of near point .

The Padé approximant of at .

Coefficient of the term in the series expansion around of the function : .

Coefficient of the term in the series expansion around of the function : .

Coefficient of the term in the series expansion around of the function : .

The residue of at the point that is equal to the coefficient of the term in the series expansion around of the function : .

The residue of at the point that is equal to the coefficient of the term in the series expansion around of the function : .

The residue of the function at the point where this function has the pole of order because :

The main factor in the coefficient of the series representation of the function through Gauss functions :

The part of the series representation of the function at the point that includes terms of the series expansions of the regular and singular components:

The asymptotic expansion of the function at the point that includes terms of the asymptotic expansions of the regular and exponential type components:

.

Infinite series or asymptotic representation of the function at the point : where means the limit of a convergent series or a Borel‐regularized infinite sum.

The nonexponential part of the asymptotic expansion (or series representation for ) of the function at the point that includes terms of each series expansion:

.

In the cases where two or more differ by integer values, the function is defined by continuity. After evaluation of the corresponding limit, the general formula includes powers of and the psi function and in such logarithmic cases the representations are very complicated.

In the following example it is assumed that no pair of the differs by an integer.

In this example, either two, three, or four differ by integers.

The exponential part of the asymptotic expansion of the function (for or ) at the point that includes terms of the series expansion:

.

The trigonometric type part of the asymptotic expansion of the function (for ) at the point that includes terms of series expansion:

.

Infinite series or the asymptotic representation of the function at the point : where means the limit of a convergent series or a Borel‐regularized infinite sum and .

In this example, it is assumed that no pair of the differs by an integer.

The following includes the most important formulas of logarithmic cases where between two and four parameters coincide or differ pairwise by an integer.

The nonexponential part of the asymptotic expansion (or series representation for ) of the function at the point that includes terms of each series expansion. In particular,

In cases where two or more differ by integer values, the function is defined by continuity. After evaluation of the corresponding limit, the general formula includes powers of and the psi function . It is too complicated for presentation here. The following formulas include the most important ones for application cases where one, two, three, or four all differ by an integer.

In the following example it is assumed that no pair of the differs by an integer.

In this example only two differ by integers.

In this example only three or four differ by integers.

The exponential part of the asymptotic expansion of the function for at the point that includes terms of series expansion:

.

The trigonometric part of the asymptotic expansion of function for at the point that includes terms of series expansion:

.

The hyperbolic part of the asymptotic expansion of the function for at the point that includes terms of series expansions:

The nonexponential part of the asymptotic expansion (or series representation for ) of the function at the point which includes terms of each series expansion. In particular,

.

In the cases where two or more differ by integer values, the function is defined by continuity. After evaluation of the corresponding limit, the general formula includes powers of and the psi function . It is too complicated for presentation here. The following formulas include the most important one for applications of cases when only two, three, or four differ by integers.

In the following example it is assumed that no pair of the differ by an integer.

In this example only two differ by integer values.

In this example only three or four differ by integers.

The exponential part of the asymptotic expansion of the function for at the point that includes terms of series expansion:

.

The trigonometric type part of the asymptotic expansion of the function for at the point that includes terms of series expansion:

.

The hyperbolic type part of the asymptotic expansion of the function for at the point that includes terms of series expansions:

Infinite series or the asymptotic representation of the function at the point : , where means the limit of a convergent series or Borel‐regularized infinite sums and .

In the following example we assume that no pair differs by an integer.

In the following example we assume that no pair differs by an integer.

In this example only two differ by integers.

In this example only two differ by integers.

In this example only three or four differ by integers.

In this example only three or four differ by integers.

The part of the series representation of the function at the point that includes terms of the series expansions of the regular and singular components, and reflects asymptotic behavior at least in the circle :

.

More detailed descriptions of

The asymptotic expansion of the function at the point that includes terms of the asymptotic expansions of the regular and exponential components:

.

.

Infinite series or asymptotic representation of the function at the point : where means the limit of a convergent series or a Borel‐regularized infinite sum.

Sum of terms : .

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 .

Double sum of over all integers except .

Double sum of terms over all integers excluding the term .

Product of terms : .

Product of excluding .

Limit of finite product .

Product over set .

Product of over all divisors of n.

Double product of function by all integers excluding the term .

Derivative of a function of argument : .

Second derivative of a function of argument : .

The derivative of a function of argument : .

The derivative of a function coincides with function : .

Partial derivative of with respect to : .

The partial derivative of with respect to : .

The general form, representing a function obtained from by differentiating times with respect to the first argument, times with respect to the second argument, and so on.

The Wronskian determinant including two functions and its derivatives: .

The Wronskian determinant for second order linear differential equation can be evaluated by the Liouville formula . The system forms a fundamental (linearly independent) set of solutions for this differential equation in a neighborhood provided does not vanish at that point.

Modified Bessel differential equations can be written in the form . So . It has solutions . Its Wronskian can be evaluated by the general formula . The Liouville formula can be used to simplify this Wronskian by selecting the corresponding point : . So .

The Wronskian determinant including functions and its derivatives: .

The Wronskian determinant for linear differential equations of the form can be evaluated by the Liouville formula . The system forms a fundamental (linearly independent) set of solutions for this differential equation in a neighborhood provided does not vanish at that point.

Fractional differentiation power constant:

.

The function is continuous for integer values of its parameters . For example, supports smooth transformation of the power function by the usual simple differentiation and integration not only into other power functions and but into functions including logarithms (when and repeated integration can be used with the corresponding negative integer ):

It is easy to verify that for integer the coefficient coincides with coefficients before after repeatable differentiation or integration of order :

The last condition can be removed if the integration is understandable in the Hadamard sense and only the finite part of the integral is taken due to the Hadamard.

For example, if and , the following evaluations can be made:

gives the logarithmic fractional differentiation constant of order with respect to : .

gives the logarithmic fractional differentiation constant of order with respect to : .

gives the logarithmic fractional differentiation constant of order with respect to : .

The fractional integro‐derivative of with respect to (which provides the Riemann‐Liouville‐Hadamard fractional left‐sided integro‐differentiation beginning at point 0):

The value is defined for analytical functions in the following way.

Suppose that the function near point can be represented through the Laurent type series

In particular for , this function is analytical near the point . In this case the value can be defined for arbitrary complex order by the formula

In particular, for

Such an approach allows the integro-derivative of fractional (generically complex) order to be defined for all functions of the hypergeometric type, including the Meijer G function, because all such functions can be represented as finite sums of the above Laurent type series.

Contour integral of function by contour .

For the bounded open contour with ranging from a to b and arbitrary points placed in order between a and b (). Thus the contour is divided into subcontours Then

For a closed contour (such as the circle ), the above procedure can be applied where on is "near" : .

For an unbounded contour with one finite end , the above procedure can be applied where on is "near" : .

For an unbounded contour with both ends infinite (such as the special contour used in the definition of the Meijer G function) define such as is divided into semi-unbounded contours and by some arbitrary finite point , such that the directions of and coincide.

Indefinite integral (antiderivative) of function . Inverse operation to differentiation: .

Definite integral of the function over interval :

Multiple definite integral of the function by the intervals .

The repeated (‐times) integral of function by interval .

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 γ-ⅈ ∞ 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 .)

Exponential Fourier integral transform of the function with respect to the variable : . (If this integral does not converge, the value of is defined in the sense of generalized functions.)

Inverse exponential Fourier integral transform of the function with respect to the variable : . (If this integral does not converge, the value of is defined in the sense of generalized functions.)

Fourier double integral transform of the function with respect to the variables , : . (If this integral does not converge, the value of is defined in the sense of generalized functions.)

Fourier cosine integral transform of the function with respect to the variable : . (If this integral does not converge, the value of is defined in the sense of generalized functions.)

Fourier sine integral transform of the function with respect to the variable : . (If this integral does not converge, the value of is defined in the sense of generalized functions.)

Hilbert transform of the function with respect to the variable : .

Hankel integral transform of the function with respect to the variable : . (If this integral does not converge, the value is defined in the sense of generalized functions.)

Laplace integral transform of the function with respect to the variable : .

Inverse Laplace integral transform of the function with respect to the variable : . (If this integral does not converge, the value of is defined in the sense of generalized functions.)

Inverse Laplace integral transform of the function with respect to the variable : . (If this integral does not converge, the value of is defined in the sense of generalized functions.)

Inverse Laplace integral transform of the function with respect to the variable : for appropriately chosen .

Laplace double integral transform of the function with respect to the variables : .

Mellin integral transform of the function with respect to the variable : . (If this integral does not converge, the value of is defined in the sense of generalized functions.)

Inverse Mellin integral transform of the function with respect to the variable : . (If this integral does not converge, the value of is defined in the sense of generalized functions.) The condition on is typically indicated in the result.

Wigner integral transform: . (If this integral does not converge, the value of is defined in the sense of generalized functions.)

The limiting value of when approaches in any direction: .

The limiting value of when approaches in direction -1: .

The limiting value of when approaches in direction 1: .

The limiting value of when approaches in direction -ⅈ: .

The limiting value of when approaches in direction ⅈ: .

The limiting value of when approaches : .

The limiting value of when approaches : .

The limiting value of when approaches : .

A finite continued fraction

Limit of the finite continued fraction .

The matrix with elements .

The determinant of the matrix with elements .





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