EllipticPi3

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

For fixed n,z

For fixed n,m

For fixed z,m

Values at infinities

General characteristics

Domain and analyticity

Π(n;z|m) is an analytical function of n, z, and m which is defined over (ℂ^3).

Symmetries and periodicities

Parity

Mirror symmetry

Periodicity

Π(n;z|m) is a quasi-periodic function with respect to z.

Quasi-symmetry

Poles and essential singularities

With respect to n

The function Π(n;z|m) does not have poles and essential singularities with respect to n.

With respect to z

The function Π(n;z|m) does not have poles and essential singularities with respect to z.

With respect to m

The function Π(n;z|m) does not have poles and essential singularities with respect to m.

Branch points

With respect to n

For fixed z, m, the function Π(n;z|m) does not have branch points.

With respect to z

With respect to m

For fixed n, z, the function Π(n;z|m) has two branch points at m==(csc^2)(z)and m==(OverTilde[∞]).

Branch cuts

With respect to n

For fixed z, m the function Π(n;z|m) does not have branch cuts.

With respect to z

For fixed m, n, the function Π(n;z|m) can have up to eight infinite sets of branch cuts (it has at least six), which form very complicated curves in the case of generic m, n.

For fixed real m, n/;max(m,n)<1, the function Π(n;z|m) does not have branch cuts on the real axis and on the vertical intervals {(csc^(-1))((Sqrt[m]))+π k,π-(csc^(-1))((Sqrt[m]))+π k}/;k∈ℤ∧m∈(-∞,1).
For fixed real m<1, the function Π(n;z|m) has six infinite sets of branch cuts, four of them are located on vertical intervals starting at the points z==π k±(csc^(-1))((Sqrt[m]))/;k∈ℤ  and extending to imaginary infinity.

1) real intervals {π k+(csc^(-1))((Sqrt[m])),π k+(π/2)}/;k∈ℤ∧m>1∧n∉(m,∞), where Π(n;z|m)  is continuous from below; if 1<n<m these intervals include two sets {π k+(csc^(-1))((Sqrt[m])),π k+(csc^(-1))((Sqrt[n]))} and {π k+(csc^(-1))((Sqrt[n])),π k+(π/2)}, where Π(n;z|m) has different values of jumps  (for generic complex m these branch cuts deform into a complicated curves; for generic complex n the intervals {π k+(csc^(-1))((Sqrt[n])),π k+(π/2)} also deform into a complicated curves);  in the case max(m,n)<1 these real intervals vanish
        2) real intervals {π k+(π/2),π (k+1)-(csc^(-1))((Sqrt[m]))}/;k∈ℤ∧m>1∧n∉(m,∞), where Π(n;z|m)  is continuous from above; if 1<n<m these intervals include two sets {π k+(π/2),π (k+1)-(csc^(-1))((Sqrt[n]))} and {π (k+1)-(csc^(-1))((Sqrt[n])),π (k+1)-(csc^(-1))((Sqrt[m]))}, where Π(n;z|m) has different values of jumps (for generic complex m these branch cuts deform into a complicated curves; for generic complex n the intervals {π k+(π/2),π (k+1)-(csc^(-1))((Sqrt[n]))} also deform into a complicated curves); in the case max(m,n)<1 these real intervals vanish
        3) real intervals {π k+(csc^(-1))((Sqrt[n])),π k+(π/2)}/;k∈ℤ∧m∉(n,∞)∧n>1, where Π(n;z|m)  is continuous from below; if 1<m<n these intervals include two sets {π k+(csc^(-1))((Sqrt[n])),π k+(csc^(-1))((Sqrt[m]))} and {π k+(csc^(-1))((Sqrt[m])),π k+(π/2)}, where Π(n;z|m) has different values of jumps (for generic complex n these branch cuts deform into a complicated curves; for generic complex m the intervals {π k+(csc^(-1))((Sqrt[m])),π k+(π/2)} also deform into a complicated curves); in the case max(m,n)<1 these real intervals vanish
        4) real intervals {π k+(π/2),π (k+1)-(csc^(-1))((Sqrt[n]))}/;k∈ℤ∧m∉(n,∞)∧n>1, where Π(n;z|m)  is continuous from above; if 1<m<n these intervals include two sets {π k+(π/2),π (k+1)-(csc^(-1))((Sqrt[m]))} and {π (k+1)-(csc^(-1))((Sqrt[m])),π (k+1)-(csc^(-1))((Sqrt[n]))}, where Π(n;z|m) has different values of jumps (for generic complex n these branch cuts deform into a complicated curves; for generic complex m the intervals {π k+(π/2),π (k+1)-(csc^(-1))((Sqrt[m]))} also deform into a complicated curves); in the case max(m,n)<1 these real intervals vanish
        5) vertical intervals {(π/2)+2π k ,(π/2)+2π k+i ∞}/;k∈ℤ∧m∉(0,1),or {π-(csc^(-1))((Sqrt[m]))+2π k ,(π/2)+2π k+i ∞}/;k∈ℤ∧m∈(0,1), where Π(n;z|m)  is continuous from the left; 
        6) vertical intervals {((3π)/2)+2π k,((3π)/2)+2π k+i ∞}/;k∈ℤ∧m∉(0,1), or {2π-(csc^(-1))((Sqrt[m]))+2π k ,((3π)/2)+2π k+i ∞}/;k∈ℤ∧m∈(0,1), where Π(n;z|m)  is continuous from the right; 
        7) vertical intervals {(π/2)+2π k-i ∞,(π/2)+2π k}/;k∈ℤ∧m∉(0,1), or {(π/2)+2π k-i ∞,2π k+(csc^(-1))((Sqrt[m]))}/;k∈ℤ∧m∈(0,1), where Π(n;z|m)  is continuous from the left; 
        8) vertical intervals {((3π)/2)+2π k-i ∞,((3π)/2)+2π k}/;k∈ℤ∧m∉(0,1), or {((3π)/2)+2π k-i ∞,2π k+π+(csc^(-1))((Sqrt[m]))}/;k∈ℤ∧m∈(0,1), where Π(n;z|m)  is continuous from the right.

For fixed real m, n/;max(m,n)<1, the function Π(n;z|m) does not have branch cuts on the real axis.

For fixed real m<1, the function Π(n;z|m) has branch points (csc^(-1))((Sqrt[m]))+π k/;k∈ℤ and π-(csc^(-1))((Sqrt[m]))+π k/;k∈ℤ. In this case branch cuts lay at the vertical lines beginning from these points and going to imaginary infinity. By this reason for fixed real m<1, the function Π(n;z|m) does not have branch cuts on the vertical intervals {(csc^(-1))((Sqrt[m]))+π k,π-(csc^(-1))((Sqrt[m]))+π k}/;k∈ℤ∧m∈(-∞,1).

With respect to m

Branch cut locations: complicated.

Series representations

Generalized power series

Expansions at generic point n==(Subscript[n, 0])

Expansions at n==0

Expansions at n==1

Expansions at n==∞

Expansions at generic point z==(Subscript[z, 0])

Expansions on branch cuts

Expansions at z==0

Expansions at z==(csc^(-1))((Sqrt[m]))+π u/;u∈ℤ

Expansions at z==-(csc^(-1))((Sqrt[m]))+π u/;u∈ℤ

Expansions at z==(csc^(-1))((Sqrt[n]))+π u/;u∈ℤ

Expansions at z==∞

Expansions at generic point m==(Subscript[m, 0])

Expansions at m==0

Expansions at m==1

Expansions at m==∞

Other series representations

Expansions EllipticPi[n,ArcSin[z],m] at z==∞

Other expansions

Integral representations

On the real axis

Of the direct function

Differential equations

Ordinary linear differential equations and wronskians

For the direct function itself

Ordinary nonlinear differential equations

Transformations

Transformations and argument simplifications

Argument involving basic arithmetic operations

Products, sums, and powers of the direct function

Sums of the direct function

Identities

Functional identities

Differentiation

Low-order differentiation

With respect to n

With respect to z

With respect to m

Symbolic differentiation

With respect to n

With respect to z

With respect to m

Fractional integro-differentiation

With respect to n

With respect to z

With respect to m

Integration

Indefinite integration

Involving only one direct function

Involving one direct function and elementary functions

Involving only one direct function with respect to n

Involving only one direct function with respect to m

Involving one direct function and elementary functions with respect to m

Representations through more general functions

Through hypergeometric functions of several variables

Through hypergeometric functions of two variables

Theorems

The length of the geodesics of an ellipsoid

The length's of the geodesics of an ellipsoid can be expressed in terms of Π(n;z|m) .

History

– A. M. Legendre (1811)

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