#REDIRECT elliptic function
In complex analysis, an elliptic function is, roughly speaking , a function (mathematics) defined on the complex plane which is periodic function in two directions. The elliptic functions can be seen as analogs of the trigonometric functions (which have a single period only). Historically, elliptic functions were discovered as inverse functions of elliptic integrals; these in turn were studied in connection with the problem of the arc length of an ellipse, whence the name derives. Formally, an elliptic function is a meromorphic function ''f'' defined on C for which there exist two non-zero complex numbers ''a'' and ''b'' such that :''f''(''z'' + ''a'') = ''f''(''z'' + ''b'') = ''f''(''z'') for all ''z'' in C and such that ''a''/''b'' is not real number. From this it follows that :''f''(''z'' + ''ma'' + ''nb'') = ''f''(''z'') for all ''z'' in C and all integers ''m'' and ''n''. In developments of the theory of elliptic functions, modern authors mostly follow Karl Weierstrass: the notations of Weierstrass's elliptic functions based on his ''''-function are convenient, and any elliptic function can be expressed in terms of these. Weierstrass became interested in these functions as a student of Christoph Gudermann, a student of Carl Friedrich Gauss. The Jacobi's elliptic functions introduced by Carl Jacobi, and the auxiliary theta functions (not doubly-periodic), are more complex; but important both for the history and for general theory. The primary difference between these two theories is that the Weierstrass functions have high-order pole (complex analysis) located at the corners of the periodic lattice (group), whereas the Jacobi functions have simple poles. The development of the Weierstrass theory is easier to present and understand, having fewer complications. Elliptic functions are the inverse functions of elliptic integrals, which is how they were introduced historically. More generally, the study of elliptic functions is closely related to the study of modular functions and modular forms, examples of which include the j-invariant, the Eisenstein series and the Dedekind eta function. ==Definition and properties== Any complex number ω such that ''f''(''z'' + ω) = ''f''(''z'') for all ''z'' in C is called a ''period'' of ''f''. If the two periods ''a'' and ''b'' are such that any other period ω can be written as ω = ''ma'' + ''nb'' with integers ''m'' and ''n'', then ''a'' and ''b'' are called ''fundamental periods''. Every elliptic function has a pair of fundamental periods, but this pair is not unique, as described below. If ''a'' and ''b'' are fundamental periods describing a lattice, then exactly the same lattice can be obtained by the fundamental periods ''a' '' and ''b' '' where ''a' '' = ''p'' ''a'' + ''q'' ''b'' and ''b' '' = ''r'' ''a'' + ''q'' ''b'' where ''p'', ''q'', ''r'' and ''s'' being integers satisfying ''p'' ''s'' - ''q'' ''r'' = 1. That is, the matrix has determinant one, and thus belongs to the modular group. In other words, if ''a'' and ''b'' are fundamental periods of an elliptic function, then so are ''a' '' and ''b' ''. If ''a'' and ''b'' are fundamental periods, then any parallelogram with vertices ''z'', ''z'' + ''a'', ''z'' + ''b'', ''z'' + ''a'' + ''b'' is called a ''fundamental parallelogram''. Shifting such a parallelogram by integral multiples of ''a'' and ''b'' yields a copy of the parallelogram, and the function ''f'' behaves identically on all these copies, because of the periodicity. The number of pole (complex analysis) in any fundamental parallelogram is finite (and the same for all fundamental parallelograms). Unless the elliptic function is constant, any fundamental parallelogram has at least one pole, a consequence of Liouville's theorem (complex analysis). The sum of the orders of the poles in any fundamental parallelogram is called the ''order'' of the elliptic function. The sum of the residue (complex analysis) of the poles in any fundamental parallelogram is equal to zero, so in particular no elliptic function can have order one. The number of zeros (counted with multiplicity) in any fundamental parallelogram is equal to the order of the elliptic function. The derivative of an elliptic function is again an elliptic function, with the same periods. The set of all elliptic functions with the same fundamental periods form a field (mathematics). The Weierstrass elliptic function is the prototypical elliptic function, and in fact, the field of elliptic functions with respect to a given lattice is generated by and its derivative . == References == * Milton Abramowitz and Irene A. Stegun, eds. ''Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables''. New York: Dover, 1972. ''(See Chapter 16.)'' * Naum Illyich Akhiezer, ''Elements of the Theory of Elliptic Functions'', (1970) Moscow, translated into English as ''AMS Translations of Mathematical Monographs Volume 79'' (1990) AMS, Rhode Island ISBN 0-8218-4532-2 * Tom M. Apostol, ''Modular Functions and Dirichlet Series in Number Theory'', Springer-Verlag, New York, 1976. ISBN 0-387-97127-0 ''(See Chapter 1.)'' * Albert Eagle, ''The elliptic functions as they should be''. Galloway and Porter, Cambridge, England 1958. * E. T. Whittaker and G. N. Watson. ''A course of modern analysis'', Cambridge University Press, 1952 Elliptic functions Modular forms Analytic number theory
==Lattice== We need an article on fundamental pair of periods that reviews all of the properties of a 2D lattice so that this article and the modular forms article (and the Jacobi & Wierestrass elliptic articles) can reference it. User:Linas 05:10, 13 Feb 2005 (UTC) :See my comment at modular form. User:Charles Matthews 08:17, 13 Feb 2005 (UTC) ==Vandalism== The page has been vandalised. User:Charles Matthews 06:03, 9 Sep 2003 (EDT) ==Weierstrass== I moved the following from the subject page: :An elliptic function on the complex numbers is a function (mathematics) of the form ::E(''z''; ''a'',''b'') = ∑''m''∑''n'' (''z'' - ''m''''a'' -''n''''b'')-2 :where ''a'' and ''b'' are complex parameters and ''m'' and ''n'' range over the integers. As written, this series is improper and divergent; but it can be made convergent by taking the Cauchy principal value, which is the limit as ''x''->∞ of the sum of those terms with |''z'' - ''m''''a'' - ''n''''b''| < ''x''. :The function is periodic with two periods, ''a'' and ''b''. Plotting E(''z'') on ''x'' versus E'(''z'') on ''y'' results in an elliptic curve. :A real number elliptic function can also be defined in the same way. Either ''a'' is real and ''b'' imaginary (in which case the elliptic curve has two parts, E(''z'' + ''b''/2) being also real for real ''z'') or ''a'' + ''b'' is real and ''a'' - ''b'' is imaginary (in which case the elliptic curve has one part). :Degenerate elliptic functions and curves are obtained by setting ''a'' or ''b'' to infinity. If ''a'' or ''b'' is infinite, but not both, the Cauchy principal value diverges and other means must be used to define the function. If both are infinite, E(''z'') is simply 1/''z''2. If ''a'' is real and ''b'' is infinite, the curve consists of one smooth part and one point. If ''a'' is imaginary and ''b'' is infinite, the curve is a loop that crosses itself. If both are infinite, the curve is the semicubical parabola ''x''3 = ''y''2/64. The formula for ''E'' is wrong I believe, and there are certainly other elliptic functions. I don't know how to rescue this. User:AxelBoldt 01:48 Nov 8, 2002 (UTC) I just picked up the yellow book. The correct formula is E(''z''; ''a'',''b'') = ''z''-2 + ∑''m''∑''n'' (''z'' - ''m''''a'' -''n''''b'')-2-(''n''''b'')-2, where ''n''=''m''=0 is excluded from the sum. I think it should be put at Weierstrass's elliptic function. -User:PierreAbbat ==References== ''The elliptic functions as they should be'' in the references is eccentric. Better for example to go to Whittaker & Watson, though their notation is not what the modern standard is (same for all the older books). Tannery and Molk is the classic reference; book by Weber. But the old books are out of print, I suppose - more's the pity. User:Charles Matthews 22:14, 19 Nov 2004 (UTC)
See other meanings of words starting from letter:
Words begining with Elliptic_function:
Elliptic_Function
Elliptic_function
Elliptic_function
Elliptic_functions
Elliptic_functions
Elliptic_functions_(Jacobi)
Elliptic Function
#REDIRECT elliptic function
Elliptic function
In complex analysis, an elliptic function is, roughly speaking , a function (mathematics) defined on the complex plane which is periodic function in two directions. The elliptic functions can be seen as analogs of the trigonometric functions (which have a single period only). Historically, elliptic functions were discovered as inverse functions of elliptic integrals; these in turn were studied in connection with the problem of the arc length of an ellipse, whence the name derives. Formally, an elliptic function is a meromorphic function ''f'' defined on C for which there exist two non-zero complex numbers ''a'' and ''b'' such that :''f''(''z'' + ''a'') = ''f''(''z'' + ''b'') = ''f''(''z'') for all ''z'' in C and such that ''a''/''b'' is not real number. From this it follows that :''f''(''z'' + ''ma'' + ''nb'') = ''f''(''z'') for all ''z'' in C and all integers ''m'' and ''n''. In developments of the theory of elliptic functions, modern authors mostly follow Karl Weierstrass: the notations of Weierstrass's elliptic functions based on his ''''-function are convenient, and any elliptic function can be expressed in terms of these. Weierstrass became interested in these functions as a student of Christoph Gudermann, a student of Carl Friedrich Gauss. The Jacobi's elliptic functions introduced by Carl Jacobi, and the auxiliary theta functions (not doubly-periodic), are more complex; but important both for the history and for general theory. The primary difference between these two theories is that the Weierstrass functions have high-order pole (complex analysis) located at the corners of the periodic lattice (group), whereas the Jacobi functions have simple poles. The development of the Weierstrass theory is easier to present and understand, having fewer complications. Elliptic functions are the inverse functions of elliptic integrals, which is how they were introduced historically. More generally, the study of elliptic functions is closely related to the study of modular functions and modular forms, examples of which include the j-invariant, the Eisenstein series and the Dedekind eta function. ==Definition and properties== Any complex number ω such that ''f''(''z'' + ω) = ''f''(''z'') for all ''z'' in C is called a ''period'' of ''f''. If the two periods ''a'' and ''b'' are such that any other period ω can be written as ω = ''ma'' + ''nb'' with integers ''m'' and ''n'', then ''a'' and ''b'' are called ''fundamental periods''. Every elliptic function has a pair of fundamental periods, but this pair is not unique, as described below. If ''a'' and ''b'' are fundamental periods describing a lattice, then exactly the same lattice can be obtained by the fundamental periods ''a' '' and ''b' '' where ''a' '' = ''p'' ''a'' + ''q'' ''b'' and ''b' '' = ''r'' ''a'' + ''q'' ''b'' where ''p'', ''q'', ''r'' and ''s'' being integers satisfying ''p'' ''s'' - ''q'' ''r'' = 1. That is, the matrix has determinant one, and thus belongs to the modular group. In other words, if ''a'' and ''b'' are fundamental periods of an elliptic function, then so are ''a' '' and ''b' ''. If ''a'' and ''b'' are fundamental periods, then any parallelogram with vertices ''z'', ''z'' + ''a'', ''z'' + ''b'', ''z'' + ''a'' + ''b'' is called a ''fundamental parallelogram''. Shifting such a parallelogram by integral multiples of ''a'' and ''b'' yields a copy of the parallelogram, and the function ''f'' behaves identically on all these copies, because of the periodicity. The number of pole (complex analysis) in any fundamental parallelogram is finite (and the same for all fundamental parallelograms). Unless the elliptic function is constant, any fundamental parallelogram has at least one pole, a consequence of Liouville's theorem (complex analysis). The sum of the orders of the poles in any fundamental parallelogram is called the ''order'' of the elliptic function. The sum of the residue (complex analysis) of the poles in any fundamental parallelogram is equal to zero, so in particular no elliptic function can have order one. The number of zeros (counted with multiplicity) in any fundamental parallelogram is equal to the order of the elliptic function. The derivative of an elliptic function is again an elliptic function, with the same periods. The set of all elliptic functions with the same fundamental periods form a field (mathematics). The Weierstrass elliptic function is the prototypical elliptic function, and in fact, the field of elliptic functions with respect to a given lattice is generated by and its derivative . == References == * Milton Abramowitz and Irene A. Stegun, eds. ''Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables''. New York: Dover, 1972. ''(See Chapter 16.)'' * Naum Illyich Akhiezer, ''Elements of the Theory of Elliptic Functions'', (1970) Moscow, translated into English as ''AMS Translations of Mathematical Monographs Volume 79'' (1990) AMS, Rhode Island ISBN 0-8218-4532-2 * Tom M. Apostol, ''Modular Functions and Dirichlet Series in Number Theory'', Springer-Verlag, New York, 1976. ISBN 0-387-97127-0 ''(See Chapter 1.)'' * Albert Eagle, ''The elliptic functions as they should be''. Galloway and Porter, Cambridge, England 1958. * E. T. Whittaker and G. N. Watson. ''A course of modern analysis'', Cambridge University Press, 1952 Elliptic functions Modular forms Analytic number theory
Elliptic function
==Lattice== We need an article on fundamental pair of periods that reviews all of the properties of a 2D lattice so that this article and the modular forms article (and the Jacobi & Wierestrass elliptic articles) can reference it. User:Linas 05:10, 13 Feb 2005 (UTC) :See my comment at modular form. User:Charles Matthews 08:17, 13 Feb 2005 (UTC) ==Vandalism== The page has been vandalised. User:Charles Matthews 06:03, 9 Sep 2003 (EDT) ==Weierstrass== I moved the following from the subject page: :An elliptic function on the complex numbers is a function (mathematics) of the form ::E(''z''; ''a'',''b'') = ∑''m''∑''n'' (''z'' - ''m''''a'' -''n''''b'')-2 :where ''a'' and ''b'' are complex parameters and ''m'' and ''n'' range over the integers. As written, this series is improper and divergent; but it can be made convergent by taking the Cauchy principal value, which is the limit as ''x''->∞ of the sum of those terms with |''z'' - ''m''''a'' - ''n''''b''| < ''x''. :The function is periodic with two periods, ''a'' and ''b''. Plotting E(''z'') on ''x'' versus E'(''z'') on ''y'' results in an elliptic curve. :A real number elliptic function can also be defined in the same way. Either ''a'' is real and ''b'' imaginary (in which case the elliptic curve has two parts, E(''z'' + ''b''/2) being also real for real ''z'') or ''a'' + ''b'' is real and ''a'' - ''b'' is imaginary (in which case the elliptic curve has one part). :Degenerate elliptic functions and curves are obtained by setting ''a'' or ''b'' to infinity. If ''a'' or ''b'' is infinite, but not both, the Cauchy principal value diverges and other means must be used to define the function. If both are infinite, E(''z'') is simply 1/''z''2. If ''a'' is real and ''b'' is infinite, the curve consists of one smooth part and one point. If ''a'' is imaginary and ''b'' is infinite, the curve is a loop that crosses itself. If both are infinite, the curve is the semicubical parabola ''x''3 = ''y''2/64. The formula for ''E'' is wrong I believe, and there are certainly other elliptic functions. I don't know how to rescue this. User:AxelBoldt 01:48 Nov 8, 2002 (UTC) I just picked up the yellow book. The correct formula is E(''z''; ''a'',''b'') = ''z''-2 + ∑''m''∑''n'' (''z'' - ''m''''a'' -''n''''b'')-2-(''n''''b'')-2, where ''n''=''m''=0 is excluded from the sum. I think it should be put at Weierstrass's elliptic function. -User:PierreAbbat ==References== ''The elliptic functions as they should be'' in the references is eccentric. Better for example to go to Whittaker & Watson, though their notation is not what the modern standard is (same for all the older books). Tannery and Molk is the classic reference; book by Weber. But the old books are out of print, I suppose - more's the pity. User:Charles Matthews 22:14, 19 Nov 2004 (UTC)
See other meanings of words starting from letter:
E
EA | EB | EC | ED | EF | EG | EH | EI | EJ | EK | EL | EM | EN | EO | EP | ER | ES | ET | EU | EW | EX | EY | EZ |Words begining with Elliptic_function:
Elliptic_Function
Elliptic_function
Elliptic_function
Elliptic_functions
Elliptic_functions
Elliptic_functions_(Jacobi)
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