The most common form of theta function is that occurring in the theory of
elliptic functions. With respect to one of the complex variables (conventionally called z), a theta function has a property expressing its behavior with respect to the addition of a period of the associated elliptic functions, making it a
quasiperiodic function. In the abstract theory this quasiperiodicity comes from the
cohomology class of a
line bundle on a complex torus, a condition of
descent.
One interpretation of theta functions when dealing with the heat equation is that "a theta function is a special function that describes the evolution of temperature on a segment domain subject to certain boundary conditions".[2]
Throughout this article, should be interpreted as (in order to resolve issues of choice of
branch).[note 1]
where q = exp(πiτ) is the
nome and η = exp(2πiz). It is a
Jacobi form. The restriction ensures that it is an absolutely convergent series. At fixed τ, this is a
Fourier series for a 1-periodic
entire function of z. Accordingly, the theta function is 1-periodic in z:
For any fixed , the function is an entire function on the complex plane, so by
Liouville's theorem, it cannot be doubly periodic in unless it is constant, and so the best we could do is to make it periodic in and quasi-periodic in . Indeed, since and , the function is unbounded, as required by Liouville's theorem.
It is in fact the most general entire function with 2 quasi-periods, in the following sense:[3]
Theorem — If is entire and nonconstant, and satisfies the functional equations
for some constant .
If , then and . If , then for some nonzero .
Auxiliary functions
The Jacobi theta function defined above is sometimes considered along with three auxiliary theta functions, in which case it is written with a double 0 subscript:
The auxiliary (or half-period) functions are defined by
This notation follows
Riemann and
Mumford;
Jacobi's original formulation was in terms of the
nomeq = eiπτ rather than τ. In Jacobi's notation the θ-functions are written:
If we set z = 0 in the above theta functions, we obtain four functions of τ only, defined on the upper half-plane. These functions are called Theta Nullwert functions, based on the German term for zero value because of the annullation of the left entry in the theta function expression. Alternatively, we obtain four functions of q only, defined on the unit disk . They are sometimes called
theta constants:[note 2]
with the
nomeq = eiπτ.
Observe that .
These can be used to define a variety of
modular forms, and to parametrize certain curves; in particular, the Jacobi identity is
Jacobi's identities describe how theta functions transform under the
modular group, which is generated by τ ↦ τ + 1 and τ ↦ −1/τ. Equations for the first transform are easily found since adding one to τ in the exponent has the same effect as adding 1/2 to z (n ≡ n2mod 2). For the second, let
Then
Theta functions in terms of the nome
Instead of expressing the Theta functions in terms of z and τ, we may express them in terms of arguments w and the
nomeq, where w = eπiz and q = eπiτ. In this form, the functions become
We see that the theta functions can also be defined in terms of w and q, without a direct reference to the exponential function. These formulas can, therefore, be used to define the Theta functions over other
fields where the exponential function might not be everywhere defined, such as fields of
p-adic numbers.
If we express the theta function in terms of the nome q = eπiτ (noting some authors instead set q = e2πiτ) and take w = eπiz then
We therefore obtain a product formula for the theta function in the form
In terms of w and q:
where ( ; )∞ is the
q-Pochhammer symbol and θ( ; ) is the
q-theta function. Expanding terms out, the Jacobi triple product can also be written
which we may also write as
This form is valid in general but clearly is of particular interest when z is real. Similar product formulas for the auxiliary theta functions are
In particular, so we may interpret them as one-parameter deformations of the periodic functions , again validating the interpretation of the theta function as the most general 2 quasi-period function.
Integral representations
The Jacobi theta functions have the following integral representations:
The Theta Nullwert function as this integral identity:
This formula was discussed in the essay Square series generating function transformations by the mathematician Maxie Schmidt from Georgia in Atlanta.
Based on this formula following three eminent examples are given:
Furthermore, the theta examples and shall be displayed:
Proper credit for most of these results goes to Ramanujan. See
Ramanujan's lost notebook and a relevant reference at
Euler function. The Ramanujan results quoted at
Euler function plus a few elementary operations give the results below, so they are either in Ramanujan's lost notebook or follow immediately from it. See also Yi (2004).[4] Define,
If the reciprocal of the
Gelfond constant is raised to the power of the reciprocal of an odd number, then the corresponding values or values can be represented in a simplified way by using the
hyperbolic lemniscatic sine:
The mathematician
Bruce Berndt found out further values[5] of the theta function:
Further values
Many values of the theta function[6] and especially of the shown phi function can be represented in terms of the gamma function:
Nome power theorems
Direct power theorems
For the transformation of the nome[7] in the theta functions these formulas can be used:
The squares of the three theta zero-value functions with the square function as the inner function are also formed in the pattern of the
Pythagorean triples according to the Jacobi Identity. Furthermore, those transformations are valid:
These formulas can be used to compute the theta values of the cube of the nome:
And the following formulas can be used to compute the theta values of the fifth power of the nome:
Transformation at the cube root of the nome
The formulas for the theta Nullwert function values from the cube root of the elliptic nome are obtained by contrasting the two real solutions of the corresponding quartic equations:
The alternating Rogers-Ramanujan continued fraction function S(q) has the following two identities:
The theta function values from the fifth root of the nome can be represented as a rational combination of the continued fractions R and S and the theta function values from the fifth power of the nome and the nome itself. The following four equations are valid for all values q between 0 and 1:
Modulus dependent theorems
Im combination with the elliptic modulus, following formulas can be displayed:
These are the formulas for the square of the elliptic nome:
And this is an efficient formula for the cube of the nome:
For all real values the now mentioned formula is valid.
And for this formula two examples shall be given:
First calculation example with the value inserted:
Second calculation example with the value inserted:
The constant represents the
Golden ratio number exactly.
Some series identities
Sums with theta function in the result
The infinite sum[8][9] of the reciprocals of
Fibonacci numbers with odd indices has this identity:
By not using the theta function expression, following identity between two sums can be formulated:
which can be shown to be invariant under substitution of s by 1 − s. The corresponding integral for z ≠ 0 is given in the article on the
Hurwitz zeta function.
Relation to the Weierstrass elliptic function
The theta function was used by Jacobi to construct (in a form adapted to easy calculation)
his elliptic functions as the quotients of the above four theta functions, and could have been used by him to construct
Weierstrass's elliptic functions also, since
where the second derivative is with respect to z and the constant c is defined so that the
Laurent expansion of ℘(z) at z = 0 has zero constant term.
Relation to the q-gamma function
The fourth theta function – and thus the others too – is intimately connected to the
Jackson q-gamma function via the relation[11]
Relations to Dedekind eta function
Let η(τ) be the
Dedekind eta function, and the argument of the theta function as the
nomeq = eπiτ. Then,
For the theta functions these integrals[13] are valid:
The final results now shown are based on the general Cauchy sum formulas.
A solution to the heat equation
The Jacobi theta function is the
fundamental solution of the one-dimensional
heat equation with spatially periodic boundary conditions.[14] Taking z = x to be real and τ = it with t real and positive, we can write
which solves the heat equation
This theta-function solution is 1-periodic in x, and as t → 0 it approaches the periodic
delta function, or
Dirac comb, in the sense of
distributions
.
General solutions of the spatially periodic initial value problem for the heat equation may be obtained by convolving the initial data at t = 0 with the theta function.
Relation to the Heisenberg group
The Jacobi theta function is invariant under the action of a discrete subgroup of the
Heisenberg group. This invariance is presented in the article on the
theta representation of the Heisenberg group.
Generalizations
If F is a
quadratic form in n variables, then the theta function associated with F is
with the sum extending over the
lattice of integers . This theta function is a
modular form of weight n/2 (on an appropriately defined subgroup) of the
modular group. In the Fourier expansion,
the numbers RF(k) are called the representation numbers of the form.
Then, given τ ∈ , the Riemann theta function is defined as
Here, z ∈ is an n-dimensional complex vector, and the superscript T denotes the
transpose. The Jacobi theta function is then a special case, with n = 1 and τ ∈ where is the
upper half-plane. One major application of the Riemann theta function is that it allows one to give explicit formulas for meromorphic functions on compact
Riemann surfaces, as well as other auxiliary objects that figure prominently in their function theory, by taking τ to be the period matrix with respect to a canonical basis for its first
homology group.
The Riemann theta converges absolutely and uniformly on compact subsets of .
The functional equation is
which holds for all vectors a, b ∈ , and for all z ∈ and τ ∈ .
In general, for all natural numbers this formula of the Euler beta function is valid:
Exemplary elliptic integrals
In the following some Elliptic Integral Singular Values[16] are derived:
The ensuing function has the following lemniscatically elliptic antiderivative:
For the value this identity appears:
This result follows from that equation chain:
The following function has the following equianharmonic elliptic antiderivative:
For the value that identity appears:
This result follows from that equation chain:
And the following function has the following elliptic antiderivative:
For the value the following identity appears:
This result follows from that equation chain:
Combination of the integral identities with the nome
The elliptic nome function has these important values:
For the proof of the correctness of these nome values, see the article
Nome (mathematics)!
On the basis of these integral identities and the above-mentioned Definition and identities to the theta functions in the same section of this article, exemplary theta zero values shall be determined now:
Partition sequences and Pochhammer products
Regular partition number sequence
The regular partition sequence itself indicates the number of ways in which a positive
integer number can be splitted into positive integer summands. For the numbers to , the associated partition numbers with all associated number partitions are listed in the following table:
Example values of P(n) and associated number partitions
The following basic definitions apply to the
pentagonal numbers and the card house numbers:
As a further application[17] one obtains a formula for the third power of the Euler product:
Strict partition number sequence
And the strict partition sequence indicates the number of ways in which such a positive integer number can be splitted into positive integer summands such that each summand appears at most once[18] and no summand value occurs repeatedly. Exactly the same sequence[19] is also generated if in the partition only odd summands are included, but these odd summands may occur more than once. Both representations for the strict partition number sequence are compared in the following table:
Example values of Q(n) and associated number partitions
If, for a given number , all partitions are set up in such a way that the summand size never increases, and all those summands that do not have a summand of the same size to the left of themselves can be marked for each partition of this type, then it will be the resulting number[21] of the marked partitions depending on by the overpartition function .
First example:
These 14 possibilities of partition markings exist for the sum 4:
Relations of the partition number sequences to each other
In the Online Encyclopedia of Integer Sequences (OEIS), the sequence of regular partition numbers is under the code A000041, the sequence of strict partitions is under the code A000009 and the sequence of superpartitions under the code A015128. All parent partitions from index are even.
The sequence of superpartitions can be written with the regular partition sequence P[22] and the strict partition sequence Q[23] can be generated like this:
In the following table of sequences of numbers, this formula should be used as an example:
Related to this property, the following combination of two series of sums can also be set up via the function ϑ01:
Notes
^See e.g.
https://dlmf.nist.gov/20.1. Note that this is, in general, not equivalent to the usual interpretation when is outside the strip . Here, denotes the principal branch of the
complex logarithm.
^Mahlburg, Karl (2004). "The overpartition function modulo small powers of 2". Discrete Mathematics. 286 (3): 263–267.
doi:
10.1016/j.disc.2004.03.014.
Ackerman, Michael (1 February 1979). "On the generating functions of certain Eisenstein series". Mathematische Annalen. 244 (1): 75–81.
doi:
10.1007/BF01420339.
S2CID120045753.
Harry Rauch with Hershel M. Farkas: Theta functions with applications to Riemann Surfaces, Williams and Wilkins, Baltimore MD 1974,
ISBN0-683-07196-3.
Charles Hermite: Sur la résolution de l'Équation du cinquiéme degré Comptes rendus, C. R. Acad. Sci. Paris, Nr. 11, March 1858.
The most common form of theta function is that occurring in the theory of
elliptic functions. With respect to one of the complex variables (conventionally called z), a theta function has a property expressing its behavior with respect to the addition of a period of the associated elliptic functions, making it a
quasiperiodic function. In the abstract theory this quasiperiodicity comes from the
cohomology class of a
line bundle on a complex torus, a condition of
descent.
One interpretation of theta functions when dealing with the heat equation is that "a theta function is a special function that describes the evolution of temperature on a segment domain subject to certain boundary conditions".[2]
Throughout this article, should be interpreted as (in order to resolve issues of choice of
branch).[note 1]
where q = exp(πiτ) is the
nome and η = exp(2πiz). It is a
Jacobi form. The restriction ensures that it is an absolutely convergent series. At fixed τ, this is a
Fourier series for a 1-periodic
entire function of z. Accordingly, the theta function is 1-periodic in z:
For any fixed , the function is an entire function on the complex plane, so by
Liouville's theorem, it cannot be doubly periodic in unless it is constant, and so the best we could do is to make it periodic in and quasi-periodic in . Indeed, since and , the function is unbounded, as required by Liouville's theorem.
It is in fact the most general entire function with 2 quasi-periods, in the following sense:[3]
Theorem — If is entire and nonconstant, and satisfies the functional equations
for some constant .
If , then and . If , then for some nonzero .
Auxiliary functions
The Jacobi theta function defined above is sometimes considered along with three auxiliary theta functions, in which case it is written with a double 0 subscript:
The auxiliary (or half-period) functions are defined by
This notation follows
Riemann and
Mumford;
Jacobi's original formulation was in terms of the
nomeq = eiπτ rather than τ. In Jacobi's notation the θ-functions are written:
If we set z = 0 in the above theta functions, we obtain four functions of τ only, defined on the upper half-plane. These functions are called Theta Nullwert functions, based on the German term for zero value because of the annullation of the left entry in the theta function expression. Alternatively, we obtain four functions of q only, defined on the unit disk . They are sometimes called
theta constants:[note 2]
with the
nomeq = eiπτ.
Observe that .
These can be used to define a variety of
modular forms, and to parametrize certain curves; in particular, the Jacobi identity is
Jacobi's identities describe how theta functions transform under the
modular group, which is generated by τ ↦ τ + 1 and τ ↦ −1/τ. Equations for the first transform are easily found since adding one to τ in the exponent has the same effect as adding 1/2 to z (n ≡ n2mod 2). For the second, let
Then
Theta functions in terms of the nome
Instead of expressing the Theta functions in terms of z and τ, we may express them in terms of arguments w and the
nomeq, where w = eπiz and q = eπiτ. In this form, the functions become
We see that the theta functions can also be defined in terms of w and q, without a direct reference to the exponential function. These formulas can, therefore, be used to define the Theta functions over other
fields where the exponential function might not be everywhere defined, such as fields of
p-adic numbers.
If we express the theta function in terms of the nome q = eπiτ (noting some authors instead set q = e2πiτ) and take w = eπiz then
We therefore obtain a product formula for the theta function in the form
In terms of w and q:
where ( ; )∞ is the
q-Pochhammer symbol and θ( ; ) is the
q-theta function. Expanding terms out, the Jacobi triple product can also be written
which we may also write as
This form is valid in general but clearly is of particular interest when z is real. Similar product formulas for the auxiliary theta functions are
In particular, so we may interpret them as one-parameter deformations of the periodic functions , again validating the interpretation of the theta function as the most general 2 quasi-period function.
Integral representations
The Jacobi theta functions have the following integral representations:
The Theta Nullwert function as this integral identity:
This formula was discussed in the essay Square series generating function transformations by the mathematician Maxie Schmidt from Georgia in Atlanta.
Based on this formula following three eminent examples are given:
Furthermore, the theta examples and shall be displayed:
Proper credit for most of these results goes to Ramanujan. See
Ramanujan's lost notebook and a relevant reference at
Euler function. The Ramanujan results quoted at
Euler function plus a few elementary operations give the results below, so they are either in Ramanujan's lost notebook or follow immediately from it. See also Yi (2004).[4] Define,
If the reciprocal of the
Gelfond constant is raised to the power of the reciprocal of an odd number, then the corresponding values or values can be represented in a simplified way by using the
hyperbolic lemniscatic sine:
The mathematician
Bruce Berndt found out further values[5] of the theta function:
Further values
Many values of the theta function[6] and especially of the shown phi function can be represented in terms of the gamma function:
Nome power theorems
Direct power theorems
For the transformation of the nome[7] in the theta functions these formulas can be used:
The squares of the three theta zero-value functions with the square function as the inner function are also formed in the pattern of the
Pythagorean triples according to the Jacobi Identity. Furthermore, those transformations are valid:
These formulas can be used to compute the theta values of the cube of the nome:
And the following formulas can be used to compute the theta values of the fifth power of the nome:
Transformation at the cube root of the nome
The formulas for the theta Nullwert function values from the cube root of the elliptic nome are obtained by contrasting the two real solutions of the corresponding quartic equations:
The alternating Rogers-Ramanujan continued fraction function S(q) has the following two identities:
The theta function values from the fifth root of the nome can be represented as a rational combination of the continued fractions R and S and the theta function values from the fifth power of the nome and the nome itself. The following four equations are valid for all values q between 0 and 1:
Modulus dependent theorems
Im combination with the elliptic modulus, following formulas can be displayed:
These are the formulas for the square of the elliptic nome:
And this is an efficient formula for the cube of the nome:
For all real values the now mentioned formula is valid.
And for this formula two examples shall be given:
First calculation example with the value inserted:
Second calculation example with the value inserted:
The constant represents the
Golden ratio number exactly.
Some series identities
Sums with theta function in the result
The infinite sum[8][9] of the reciprocals of
Fibonacci numbers with odd indices has this identity:
By not using the theta function expression, following identity between two sums can be formulated:
which can be shown to be invariant under substitution of s by 1 − s. The corresponding integral for z ≠ 0 is given in the article on the
Hurwitz zeta function.
Relation to the Weierstrass elliptic function
The theta function was used by Jacobi to construct (in a form adapted to easy calculation)
his elliptic functions as the quotients of the above four theta functions, and could have been used by him to construct
Weierstrass's elliptic functions also, since
where the second derivative is with respect to z and the constant c is defined so that the
Laurent expansion of ℘(z) at z = 0 has zero constant term.
Relation to the q-gamma function
The fourth theta function – and thus the others too – is intimately connected to the
Jackson q-gamma function via the relation[11]
Relations to Dedekind eta function
Let η(τ) be the
Dedekind eta function, and the argument of the theta function as the
nomeq = eπiτ. Then,
For the theta functions these integrals[13] are valid:
The final results now shown are based on the general Cauchy sum formulas.
A solution to the heat equation
The Jacobi theta function is the
fundamental solution of the one-dimensional
heat equation with spatially periodic boundary conditions.[14] Taking z = x to be real and τ = it with t real and positive, we can write
which solves the heat equation
This theta-function solution is 1-periodic in x, and as t → 0 it approaches the periodic
delta function, or
Dirac comb, in the sense of
distributions
.
General solutions of the spatially periodic initial value problem for the heat equation may be obtained by convolving the initial data at t = 0 with the theta function.
Relation to the Heisenberg group
The Jacobi theta function is invariant under the action of a discrete subgroup of the
Heisenberg group. This invariance is presented in the article on the
theta representation of the Heisenberg group.
Generalizations
If F is a
quadratic form in n variables, then the theta function associated with F is
with the sum extending over the
lattice of integers . This theta function is a
modular form of weight n/2 (on an appropriately defined subgroup) of the
modular group. In the Fourier expansion,
the numbers RF(k) are called the representation numbers of the form.
Then, given τ ∈ , the Riemann theta function is defined as
Here, z ∈ is an n-dimensional complex vector, and the superscript T denotes the
transpose. The Jacobi theta function is then a special case, with n = 1 and τ ∈ where is the
upper half-plane. One major application of the Riemann theta function is that it allows one to give explicit formulas for meromorphic functions on compact
Riemann surfaces, as well as other auxiliary objects that figure prominently in their function theory, by taking τ to be the period matrix with respect to a canonical basis for its first
homology group.
The Riemann theta converges absolutely and uniformly on compact subsets of .
The functional equation is
which holds for all vectors a, b ∈ , and for all z ∈ and τ ∈ .
In general, for all natural numbers this formula of the Euler beta function is valid:
Exemplary elliptic integrals
In the following some Elliptic Integral Singular Values[16] are derived:
The ensuing function has the following lemniscatically elliptic antiderivative:
For the value this identity appears:
This result follows from that equation chain:
The following function has the following equianharmonic elliptic antiderivative:
For the value that identity appears:
This result follows from that equation chain:
And the following function has the following elliptic antiderivative:
For the value the following identity appears:
This result follows from that equation chain:
Combination of the integral identities with the nome
The elliptic nome function has these important values:
For the proof of the correctness of these nome values, see the article
Nome (mathematics)!
On the basis of these integral identities and the above-mentioned Definition and identities to the theta functions in the same section of this article, exemplary theta zero values shall be determined now:
Partition sequences and Pochhammer products
Regular partition number sequence
The regular partition sequence itself indicates the number of ways in which a positive
integer number can be splitted into positive integer summands. For the numbers to , the associated partition numbers with all associated number partitions are listed in the following table:
Example values of P(n) and associated number partitions
The following basic definitions apply to the
pentagonal numbers and the card house numbers:
As a further application[17] one obtains a formula for the third power of the Euler product:
Strict partition number sequence
And the strict partition sequence indicates the number of ways in which such a positive integer number can be splitted into positive integer summands such that each summand appears at most once[18] and no summand value occurs repeatedly. Exactly the same sequence[19] is also generated if in the partition only odd summands are included, but these odd summands may occur more than once. Both representations for the strict partition number sequence are compared in the following table:
Example values of Q(n) and associated number partitions
If, for a given number , all partitions are set up in such a way that the summand size never increases, and all those summands that do not have a summand of the same size to the left of themselves can be marked for each partition of this type, then it will be the resulting number[21] of the marked partitions depending on by the overpartition function .
First example:
These 14 possibilities of partition markings exist for the sum 4:
Relations of the partition number sequences to each other
In the Online Encyclopedia of Integer Sequences (OEIS), the sequence of regular partition numbers is under the code A000041, the sequence of strict partitions is under the code A000009 and the sequence of superpartitions under the code A015128. All parent partitions from index are even.
The sequence of superpartitions can be written with the regular partition sequence P[22] and the strict partition sequence Q[23] can be generated like this:
In the following table of sequences of numbers, this formula should be used as an example:
Related to this property, the following combination of two series of sums can also be set up via the function ϑ01:
Notes
^See e.g.
https://dlmf.nist.gov/20.1. Note that this is, in general, not equivalent to the usual interpretation when is outside the strip . Here, denotes the principal branch of the
complex logarithm.
^Mahlburg, Karl (2004). "The overpartition function modulo small powers of 2". Discrete Mathematics. 286 (3): 263–267.
doi:
10.1016/j.disc.2004.03.014.
Ackerman, Michael (1 February 1979). "On the generating functions of certain Eisenstein series". Mathematische Annalen. 244 (1): 75–81.
doi:
10.1007/BF01420339.
S2CID120045753.
Harry Rauch with Hershel M. Farkas: Theta functions with applications to Riemann Surfaces, Williams and Wilkins, Baltimore MD 1974,
ISBN0-683-07196-3.
Charles Hermite: Sur la résolution de l'Équation du cinquiéme degré Comptes rendus, C. R. Acad. Sci. Paris, Nr. 11, March 1858.