{"history":[],"article":{"id":"27c3107a-7ac8-4bc7-a18b-bfb07d0f2f07","title":"Jacobi elliptic functions","slug":"jacobi-elliptic-functions","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), the **Jacobi [elliptic functions](https://bliptext.com/articles/elliptic-function)** are a set of basic [elliptic functions](https://bliptext.com/articles/elliptic-function).\nThey are found in the description of the [motion of a pendulum](https://bliptext.com/articles/pendulum-mechanics), as well as in the design of electronic [elliptic filters](https://bliptext.com/articles/elliptic-filter).\nWhile [trigonometric functions](https://bliptext.com/articles/trigonometry) are defined with reference to a circle, the Jacobi elliptic functions are a generalization which refer to other [conic sections](https://bliptext.com/articles/conic-section), the ellipse in particular.\nThe relation to trigonometric functions is contained in the notation, for example, by the matching notation for \\sin.\nThe Jacobi elliptic functions are used more often in practical problems than the [Weierstrass elliptic functions](https://bliptext.com/articles/weierstrass-elliptic-functions) as they do not require notions of complex analysis to be defined and/or understood.\nThey were introduced by.\n[Carl Friedrich Gauss](https://bliptext.com/articles/carl-friedrich-gauss) had already studied special Jacobi elliptic functions in 1797, the [lemniscate elliptic functions](https://bliptext.com/articles/lemniscate-elliptic-functions) in particular, but his work was published much later.\n\n# Overview\nThere are twelve Jacobi elliptic functions denoted by, where \\mathrm p and \\mathrm q are any of the letters \\mathrm c, \\mathrm s, \\mathrm n, and \\mathrm d. (Functions of the form are trivially set to unity for notational completeness.) u is the argument, and m is the parameter, both of which may be complex.\nIn fact, the Jacobi elliptic functions are [meromorphic](https://bliptext.com/articles/meromorphic-function) in both u and m. The distribution of the zeros and poles in the u-plane is well-known.\nHowever, questions of the distribution of the zeros and poles in the m-plane remain to be investigated.\nIn the complex plane of the argument u, the twelve functions form a repeating lattice of simple [poles and zeroes](https://bliptext.com/articles/zeros-and-poles).\nDepending on the function, one repeating parallelogram, or unit cell, will have sides of length 2K or 4K on the real axis, and 2K' or 4K' on the imaginary axis, where K=K(m) and K'=K(1-m) are known as the [quarter periods](https://bliptext.com/articles/quarter-period) with K(\\cdot) being the [elliptic integral](https://bliptext.com/articles/elliptic-integral) of the first kind.\nThe nature of the unit cell can be determined by inspecting the \"auxiliary rectangle\" (generally a parallelogram), which is a rectangle formed by the origin (0,0) at one corner, and (K,K') as the diagonally opposite corner.\nAs in the diagram, the four corners of the auxiliary rectangle are named \\mathrm s, \\mathrm c, \\mathrm d, and \\mathrm n, going counter-clockwise from the origin.\nThe function will have a zero at the \\mathrm p corner and a pole at the \\mathrm q corner.\nThe twelve functions correspond to the twelve ways of arranging these poles and zeroes in the corners of the rectangle.\nWhen the argument u and parameter m are real, with 0 < m < 1, K and K' will be real and the auxiliary parallelogram will in fact be a rectangle, and the Jacobi elliptic functions will all be real valued on the real line.\nSince the Jacobian elliptic functions are doubly periodic in u, they factor through a [torus](https://bliptext.com/articles/torus) – in effect, their domain can be taken to be a [torus](https://bliptext.com/articles/torus), just as cosine and sine are in effect defined on a circle.\nInstead of having only one circle, we now have the product of two circles, one real and the other imaginary.\nThe complex plane can be replaced by a [complex torus](https://bliptext.com/articles/complex-torus).\nThe circumference of the first circle is 4K and the second 4K', where K and K' are the [quarter periods](https://bliptext.com/articles/quarter-period).\nEach function has two zeroes and two poles at opposite positions on the torus.\nAmong the points 0, K, K + iK', iK' there is one zero and one pole.\nThe Jacobian elliptic functions are then doubly periodic, meromorphic functions satisfying the following properties:\n# [The fundamental rectangle in the complex plane of u | upload.wikimedia.org/wikipedia/commons/9/95/JacobiFunctionAbstract.png]\n\n# Notation\nThe elliptic functions can be given in a variety of notations, which can make the subject unnecessarily confusing.\nElliptic functions are functions of two variables.\nThe first variable might be given in terms of the **amplitude** \\varphi, or more commonly, in terms of u given below.\nThe second variable might be given in terms of the **parameter** m, or as the **[elliptic modulus](https://bliptext.com/articles/elliptic-modulus)** k, where k^2=m, or in terms of the **[modular angle](https://bliptext.com/articles/modular-angle)** \\alpha, where.\nThe complements of k and m are defined as m'=1-m and.\nThese four terms are used below without comment to simplify various expressions.\nThe twelve Jacobi elliptic functions are generally written as where \\mathrm p and \\mathrm q are any of the letters \\mathrm c, \\mathrm s, \\mathrm n, and \\mathrm d. Functions of the form are trivially set to unity for notational completeness.\nThe “major” functions are generally taken to be, and from which all other functions can be derived and expressions are often written solely in terms of these three functions, however, various symmetries and generalizations are often most conveniently expressed using the full set.\n(This notation is due to [Gudermann](https://bliptext.com/articles/christof-gudermann) and [Glaisher](https://bliptext.com/articles/james-whitbread-lee-glaisher) and is not Jacobi's original notation.)\nThroughout this article,.\nThe functions are notationally related to each other by the multiplication rule: (arguments suppressed)\nfrom which other commonly used relationships can be derived:\nThe multiplication rule follows immediately from the identification of the elliptic functions with the [Neville theta functions](https://bliptext.com/articles/neville-theta-function)\nAlso note that:\n\n# Definition in terms of inverses of elliptic integrals\nThere is a definition, relating the elliptic functions to the inverse of the [incomplete elliptic integral of the first kind](https://bliptext.com/articles/elliptic-integral) F. These functions take the parameters u and m as inputs.\nThe \\varphi that satisfies\nis called the **Jacobi amplitude**:\nIn this framework, the elliptic sine sn u (Latin: sinus amplitudinis) is given by\nand the elliptic cosine cn u (Latin: cosinus amplitudinis) is given by\nand the delta amplitude dn u (Latin: delta amplitudinis)\nIn the above, the value m is a free parameter, usually taken to be real such that (but can be complex in general), and so the elliptic functions can be thought of as being given by two variables, u and the parameter m. The remaining nine elliptic functions are easily built from the above three, and are given in a section below.\nNote that when, that u then equals the [quarter period](https://bliptext.com/articles/quarter-period) K.\nIn the most general setting, is a [multivalued function](https://bliptext.com/articles/multivalued-function) (in u) with infinitely many [logarithmic branch points](https://bliptext.com/articles/branch-point) (the branches differ by integer multiples of 2\\pi), namely the points and where.\nThis multivalued function can be made single-valued by cutting the complex plane along the line segments joining these branch points (the cutting can be done in non-equivalent ways, giving non-equivalent single-valued functions), thus making [analytic](https://bliptext.com/articles/analytic-function) everywhere except on the [branch cuts](https://bliptext.com/articles/branch-point).\nIn contrast, and other elliptic functions have no branch points, give consistent values for every branch of, and are [meromorphic](https://bliptext.com/articles/meromorphic-function) in the whole complex plane.\nSince every elliptic function is meromorphic in the whole complex plane (by definition), (when considered as a single-valued function) is not an elliptic function.\nHowever, a particular cutting for can be made in the u-plane by line segments from to with ; then it only remains to define at the branch cuts by continuity from some direction.\nThen becomes single-valued and singly-periodic in u with the minimal period 4iK(1-m) and it has singularities at the logarithmic branch points mentioned above.\nIf and m\\le 1, is continuous in u on the real line.\nWhen m>1, the branch cuts of in the u-plane cross the real line at for ; therefore for m>1, is not continuous in u on the real line and jumps by 2\\pi on the discontinuities.\nBut defining this way gives rise to very complicated branch cuts in the m-plane (not the u-plane); they have not been fully described as of yet.\nLet\nbe the [incomplete elliptic integral of the second kind](https://bliptext.com/articles/elliptic-integral) with parameter m.\nThen the **Jacobi epsilon** function can be defined as for and 0 1, the curve is a hyperbola.\nWhen m is complex but not real, x or y or both are complex and the curve cannot be described on a real x-y diagram.\n# [Plot of the Jacobi hyperbola (x2 + y2/b2 = 1, b imaginary) and the twelve Jacobi Elliptic functions pq(u,m) for particular values of angle φ and parameter b. The solid curve is the hyperbola, with m = 1 − 1/b2 and u = F(φ,m) where F(⋅,⋅) is the elliptic integral of the first kind. The dotted curve is the unit circle. For the ds-dc triangle, σ = sin(φ)cos(φ). | upload.wikimedia.org/wikipedia/commons/3/3e/Jacobi///Elliptic///Functions///%28on///Jacobi///Hyperbola%29.svg]\n\n# Minor functions\nReversing the order of the two letters of the function name results in the reciprocals of the three functions above:\nSimilarly, the ratios of the three primary functions correspond to the first letter of the numerator followed by the first letter of the denominator:\nMore compactly, we have\nwhere p and q are any of the letters s, c, d.\n\n# Periodicity, poles, and residues\nIn the complex plane of the argument u, the Jacobi elliptic functions form a repeating pattern of poles (and zeroes).\nThe residues of the poles all have the same absolute value, differing only in sign.\nEach function pq(u,m) has an \"inverse function\" (in the multiplicative sense) qp(u,m) in which the positions of the poles and zeroes are exchanged.\nThe periods of repetition are generally different in the real and imaginary directions, hence the use of the term \"doubly periodic\" to describe them.\nFor the Jacobi amplitude and the Jacobi epsilon function:\nwhere E(m) is the [complete elliptic integral of the second kind](https://bliptext.com/articles/elliptic-integral) with parameter m.\nThe double periodicity of the Jacobi elliptic functions may be expressed as:\nwhere α and β are any pair of integers.\nK(⋅) is the complete elliptic integral of the first kind, also known as the [quarter period](https://bliptext.com/articles/quarter-period).\nThe power of negative unity (γ) is given in the following table:\n!colspan=\"2\" rowspan=\"2\"|\n!colspan=\"4\"|q\n! c ! s ! n ! d !rowspan=\"6\"|p\n! c ! s ! n ! d\nWhen the factor (−1)γ is equal to −1, the equation expresses quasi-periodicity.\nWhen it is equal to unity, it expresses full periodicity.\nIt can be seen, for example, that for the entries containing only α when α is even, full periodicity is expressed by the above equation, and the function has full periods of 4K(m) and 2iK(1 − m).\nLikewise, functions with entries containing only β have full periods of 2K(m) and 4iK(1 − m), while those with α + β have full periods of 4K(m) and 4iK(1 − m).\nIn the diagram on the right, which plots one repeating unit for each function, indicating phase along with the location of poles and zeroes, a number of regularities can be noted: The inverse of each function is opposite the diagonal, and has the same size unit cell, with poles and zeroes exchanged.\nThe pole and zero arrangement in the auxiliary rectangle formed by (0,0), (K,0), (0,K′) and (K,K′) are in accordance with the description of the pole and zero placement described in the introduction above.\nAlso, the size of the white ovals indicating poles are a rough measure of the absolute value of the residue for that pole.\nThe residues of the poles closest to the origin in the figure (i.e. in the auxiliary rectangle) are listed in the following table:\n!colspan=\"2\" rowspan=\"2\"|\n!colspan=\"4\"|q\n! width=\"40pt\"|c\n! width=\"40pt\"|s\n! width=\"40pt\"|n\n! width=\"40pt\"|d\n!rowspan=\"6\"|p\n! height=\"40pt\" |c\n! height=\"40pt\" |s\n! height=\"40pt\" |n\n! height=\"40pt\" | d\nWhen applicable, poles displaced above by 2K or displaced to the right by 2K′ have the same value but with signs reversed, while those diagonally opposite have the same value.\nNote that poles and zeroes on the left and lower edges are considered part of the unit cell, while those on the upper and right edges are not.\nThe information about poles can in fact be used to [characterize](https://bliptext.com/articles/characterization-mathematics) the Jacobi elliptic functions:\nThe function is the unique elliptic function having simple poles at (with ) with residues taking the value 0 at 0.\nThe function is the unique elliptic function having simple poles at (with ) with residues taking the value 1 at 0.\nThe function is the unique elliptic function having simple poles at (with ) with residues (-1)^{s-1}i taking the value 1 at 0.\n# [Plots of the phase for the twelve Jacobi Elliptic functions pq(u,m) as a function complex argument u, with poles and zeroes indicated. The plots are over one full cycle in the real and imaginary directions with the colored portion indicating phase according to the color wheel at the lower right (which replaces the trivial dd function). Regions with absolute value below 1/3 are colored black, roughly indicating the location of a zero, while regions with absolute value above 3 are colored white, roughly indicating the position of a pole. All plots use m = 2/3 with K = K(m), K′ = K(1 − m), K(⋅) being the complete elliptic integral of the first kind. Arrows at the poles point in direction of zero phase. Right and left arrows imply positive and negative real residues respectively. Up and down arrows imply positive and negative imaginary residues respectively. | upload.wikimedia.org/wikipedia/commons/a/ad/JacobiEllipticFunctions.svg]\n\n# Special values\nSetting m=-1 gives the [lemniscate elliptic functions](https://bliptext.com/articles/lemniscate-elliptic-functions) and :\nWhen m=0 or m=1, the Jacobi elliptic functions are reduced to non-elliptic functions:\nFor the Jacobi amplitude, and where is the [Gudermannian function](https://bliptext.com/articles/gudermannian-function).\nIn general if neither of p,q is d then.\n\n# Identities\n\n## Half angle formula\n\n## K formulas\n**Half K formula**\n**Third K formula**\nTo get x3, we take the tangent of twice the arctangent of the modulus.\nAlso this equation leads to the sn-value of the third of K:\nThese equations lead to the other values of the Jacobi-Functions:\n**Fifth K formula**\nFollowing equation has following solution:\nTo get the sn-values, we put the solution x into following expressions:\n\n## Relations between squares of the functions\nRelations between squares of the functions can be derived from two basic relationships (Arguments (u,m) suppressed):\nwhere m + m' = 1.\nMultiplying by any function of the form nq yields more general equations:\nWith q = d, these correspond trigonometrically to the equations for the unit circle (x^2+y^2=r^2) and the unit ellipse, with x = cd, y = sd and r = nd.\nUsing the multiplication rule, other relationships may be derived.\nFor example:\n\n## Addition theorems\nThe functions satisfy the two square relations (dependence on m suppressed)\nFrom this we see that (cn, sn, dn) parametrizes an [elliptic curve](https://bliptext.com/articles/elliptic-curve) which is the intersection of the two [quadrics](https://bliptext.com/articles/quadric) defined by the above two equations.\nWe now may define a group law for points on this curve by the addition formulas for the Jacobi functions\nThe Jacobi epsilon and zn functions satisfy a quasi-addition theorem:\nDouble angle formulae can be easily derived from the above equations by setting x = y.\nHalf angle formulae are all of the form:\nwhere:\n\n# Jacobi elliptic functions as solutions of nonlinear ordinary differential equations\n\n## Derivatives with respect to the first variable\nThe [derivatives](https://bliptext.com/articles/derivative) of the three basic Jacobi elliptic functions (with respect to the first variable, with m fixed) are:\nThese can be used to derive the derivatives of all other functions as shown in the table below (arguments (u,m) suppressed):\nAlso\nWith the [addition theorems above](https://bliptext.com/articles/) and for a given m with 0 < m < 1 the major functions are therefore solutions to the following nonlinear [ordinary differential equations](https://bliptext.com/articles/ordinary-differential-equation):\nThe function which exactly solves the [pendulum differential equation](https://bliptext.com/articles/pendulum-mechanics),\nwith initial angle \\theta_0 and zero initial angular velocity is\nwhere, c>0 and.\n\n## Derivatives with respect to the second variable\nWith the first argument z fixed, the derivatives with respect to the second variable m are as follows:\n\n# Expansion in terms of the nome\nLet the [nome](https://bliptext.com/articles/nome-mathematics) be, , m=k^2 and let.\nThen the functions have expansions as [Lambert series](https://bliptext.com/articles/lambert-series)\nwhen\nBivariate power series expansions have been published by Schett.\n\n# Fast computation\nThe theta function ratios provide an efficient way of computing the Jacobi elliptic functions.\nThere is an alternative method, based on the [arithmetic-geometric mean](https://bliptext.com/articles/arithmetic-geometric-mean) and [Landen's transformations](https://bliptext.com/articles/landen-s-transformation):\nInitialize\nwhere 01, is not continuous in u on the real line and jumps by 2\\pi on the discontinuities.\nBut defining this way gives rise to very complicated branch cuts in the m-plane (not the u-plane); they have not been fully described as of yet.\nLet\nbe the [incomplete elliptic integral of the second kind](https://bliptext.com/articles/elliptic-integral) with parameter m.\nThen the **Jacobi epsilon** function can be defined as for and 0 1, the curve is a hyperbola.\nWhen m is complex but not real, x or y or both are complex and the curve cannot be described on a real x-y diagram.\n# [Plot of the Jacobi hyperbola (x2 + y2/b2 = 1, b imaginary) and the twelve Jacobi Elliptic functions pq(u,m) for particular values of angle φ and parameter b. The solid curve is the hyperbola, with m = 1 − 1/b2 and u = F(φ,m) where F(⋅,⋅) is the elliptic integral of the first kind. The dotted curve is the unit circle. For the ds-dc triangle, σ = sin(φ)cos(φ). | upload.wikimedia.org/wikipedia/commons/3/3e/Jacobi///Elliptic///Functions///%28on///Jacobi///Hyperbola%29.svg]\n\n# Minor functions\nReversing the order of the two letters of the function name results in the reciprocals of the three functions above:\nSimilarly, the ratios of the three primary functions correspond to the first letter of the numerator followed by the first letter of the denominator:\nMore compactly, we have\nwhere p and q are any of the letters s, c, d.\n\n# Periodicity, poles, and residues\nIn the complex plane of the argument u, the Jacobi elliptic functions form a repeating pattern of poles (and zeroes).\nThe residues of the poles all have the same absolute value, differing only in sign.\nEach function pq(u,m) has an \"inverse function\" (in the multiplicative sense) qp(u,m) in which the positions of the poles and zeroes are exchanged.\nThe periods of repetition are generally different in the real and imaginary directions, hence the use of the term \"doubly periodic\" to describe them.\nFor the Jacobi amplitude and the Jacobi epsilon function:\nwhere E(m) is the [complete elliptic integral of the second kind](https://bliptext.com/articles/elliptic-integral) with parameter m.\nThe double periodicity of the Jacobi elliptic functions may be expressed as:\nwhere α and β are any pair of integers.\nK(⋅) is the complete elliptic integral of the first kind, also known as the [quarter period](https://bliptext.com/articles/quarter-period).\nThe power of negative unity (γ) is given in the following table:\n!colspan=\"2\" rowspan=\"2\"|\n!colspan=\"4\"|q\n! c ! s ! n ! d !rowspan=\"6\"|p\n! c ! s ! n ! d\nWhen the factor (−1)γ is equal to −1, the equation expresses quasi-periodicity.\nWhen it is equal to unity, it expresses full periodicity.\nIt can be seen, for example, that for the entries containing only α when α is even, full periodicity is expressed by the above equation, and the function has full periods of 4K(m) and 2iK(1 − m).\nLikewise, functions with entries containing only β have full periods of 2K(m) and 4iK(1 − m), while those with α + β have full periods of 4K(m) and 4iK(1 − m).\nIn the diagram on the right, which plots one repeating unit for each function, indicating phase along with the location of poles and zeroes, a number of regularities can be noted: The inverse of each function is opposite the diagonal, and has the same size unit cell, with poles and zeroes exchanged.\nThe pole and zero arrangement in the auxiliary rectangle formed by (0,0), (K,0), (0,K′) and (K,K′) are in accordance with the description of the pole and zero placement described in the introduction above.\nAlso, the size of the white ovals indicating poles are a rough measure of the absolute value of the residue for that pole.\nThe residues of the poles closest to the origin in the figure (i.e. in the auxiliary rectangle) are listed in the following table:\n!colspan=\"2\" rowspan=\"2\"|\n!colspan=\"4\"|q\n! width=\"40pt\"|c\n! width=\"40pt\"|s\n! width=\"40pt\"|n\n! width=\"40pt\"|d\n!rowspan=\"6\"|p\n! height=\"40pt\" |c\n! height=\"40pt\" |s\n! height=\"40pt\" |n\n! height=\"40pt\" | d\nWhen applicable, poles displaced above by 2K or displaced to the right by 2K′ have the same value but with signs reversed, while those diagonally opposite have the same value.\nNote that poles and zeroes on the left and lower edges are considered part of the unit cell, while those on the upper and right edges are not.\nThe information about poles can in fact be used to [characterize](https://bliptext.com/articles/characterization-mathematics) the Jacobi elliptic functions:\nThe function is the unique elliptic function having simple poles at (with ) with residues taking the value 0 at 0.\nThe function is the unique elliptic function having simple poles at (with ) with residues taking the value 1 at 0.\nThe function is the unique elliptic function having simple poles at (with ) with residues (-1)^{s-1}i taking the value 1 at 0.\n# [Plots of the phase for the twelve Jacobi Elliptic functions pq(u,m) as a function complex argument u, with poles and zeroes indicated. The plots are over one full cycle in the real and imaginary directions with the colored portion indicating phase according to the color wheel at the lower right (which replaces the trivial dd function). Regions with absolute value below 1/3 are colored black, roughly indicating the location of a zero, while regions with absolute value above 3 are colored white, roughly indicating the position of a pole. All plots use m = 2/3 with K = K(m), K′ = K(1 − m), K(⋅) being the complete elliptic integral of the first kind. Arrows at the poles point in direction of zero phase. Right and left arrows imply positive and negative real residues respectively. Up and down arrows imply positive and negative imaginary residues respectively. | upload.wikimedia.org/wikipedia/commons/a/ad/JacobiEllipticFunctions.svg]\n\n# Special values\nSetting m=-1 gives the [lemniscate elliptic functions](https://bliptext.com/articles/lemniscate-elliptic-functions) and :\nWhen m=0 or m=1, the Jacobi elliptic functions are reduced to non-elliptic functions:\nFor the Jacobi amplitude, and where is the [Gudermannian function](https://bliptext.com/articles/gudermannian-function).\nIn general if neither of p,q is d then.\n\n# Identities\n\n## Half angle formula\n\n## K formulas\n**Half K formula**\n**Third K formula**\nTo get x3, we take the tangent of twice the arctangent of the modulus.\nAlso this equation leads to the sn-value of the third of K:\nThese equations lead to the other values of the Jacobi-Functions:\n**Fifth K formula**\nFollowing equation has following solution:\nTo get the sn-values, we put the solution x into following expressions:\n\n## Relations between squares of the functions\nRelations between squares of the functions can be derived from two basic relationships (Arguments (u,m) suppressed):\nwhere m + m' = 1.\nMultiplying by any function of the form nq yields more general equations:\nWith q = d, these correspond trigonometrically to the equations for the unit circle (x^2+y^2=r^2) and the unit ellipse, with x = cd, y = sd and r = nd.\nUsing the multiplication rule, other relationships may be derived.\nFor example:\n\n## Addition theorems\nThe functions satisfy the two square relations (dependence on m suppressed)\nFrom this we see that (cn, sn, dn) parametrizes an [elliptic curve](https://bliptext.com/articles/elliptic-curve) which is the intersection of the two [quadrics](https://bliptext.com/articles/quadric) defined by the above two equations.\nWe now may define a group law for points on this curve by the addition formulas for the Jacobi functions\nThe Jacobi epsilon and zn functions satisfy a quasi-addition theorem:\nDouble angle formulae can be easily derived from the above equations by setting x = y.\nHalf angle formulae are all of the form:\nwhere:\n\n# Jacobi elliptic functions as solutions of nonlinear ordinary differential equations\n\n## Derivatives with respect to the first variable\nThe [derivatives](https://bliptext.com/articles/derivative) of the three basic Jacobi elliptic functions (with respect to the first variable, with m fixed) are:\nThese can be used to derive the derivatives of all other functions as shown in the table below (arguments (u,m) suppressed):\nAlso\nWith the [addition theorems above](https://bliptext.com/articles/) and for a given m with 0 0 and.\n\n## Derivatives with respect to the second variable\nWith the first argument z fixed, the derivatives with respect to the second variable m are as follows:\n\n# Expansion in terms of the nome\nLet the [nome](https://bliptext.com/articles/nome-mathematics) be, , m=k^2 and let.\nThen the functions have expansions as [Lambert series](https://bliptext.com/articles/lambert-series)\nwhen\nBivariate power series expansions have been published by Schett.\n\n# Fast computation\nThe theta function ratios provide an efficient way of computing the Jacobi elliptic functions.\nThere is an alternative method, based on the [arithmetic-geometric mean](https://bliptext.com/articles/arithmetic-geometric-mean) and [Landen's transformations](https://bliptext.com/articles/landen-s-transformation):\nInitialize\nwhere 0

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