where a < sub > 0 </ sub >, ..., a < sub > p − 1 </ sub > are integers and ζ is a complex number such that

where, by definition, the left hand side is ζ ( s ) and the infinite product on the right hand side extends over all prime numbers p ( such expressions are called Euler products ):

: where ζ ( s ) is the Riemann zeta function.

where ζ is the Riemann zeta function and 1 / ζ ( 2 ) is approximately 0. 6079 ( over 3 / 5 of the integers are square-free ).

Let φ ( ξ, η, ζ ) be an arbitrary function of three independent variables, and let the spherical wave form F be a delta-function: that is, let F be a weak limit of continuous functions whose integral is unity, but whose support ( the region where the function is non-zero ) shrinks to the origin.

In cases where the binary star has a Bayer designation and is widely separated, it is possible that the members of the pair will be designated with superscripts ; an example is ζ Reticuli, whose components are ζ < sup > 1 </ sup > Reticuli and ζ < sup > 2 </ sup > Reticuli.

where ζ ( s ) is the Riemann zeta function ( which is undefined for s = 1 ).

where ζ ( s, q ) is the Hurwitz zeta ; thus, in a certain sense, the Hurwitz zeta generalizes the Bernoulli polynomials to non-integer values of n.

More precisely, ζ ( X, s ) can be written as a finite alternating product < dl >< dd ></ dl > where each P < sub > i </ sub >( T ) is an integral polynomial.

where ε < sub > r </ sub > is the dielectric constant of the dispersion medium, ε < sub > 0 </ sub > is the permittivity of free space ( C² N < sup >− 1 </ sup > m < sup >− 2 </ sup >), η is dynamic viscosity of the dispersion medium ( Pa s ), and ζ is zeta potential ( i. e., the electrokinetic potential of the slipping plane in the double layer ).

Fixing an integer k ≥ 1, the Dirichlet L-functions for characters modulo k are linear combinations, with constant coefficients, of the ζ ( s, q ) where q

where ζ here is the Riemann zeta function.

where N is an integer, held fixed, and ζ ( s, n ) is the Hurwitz zeta function.

where q = exp ( πiτ ) and ζ = exp ( 2πiz ).

where p runs over all primes, ζ ( s ) denotes the Riemann zeta function, and ζ ( 3 ) is Apéry's constant ( Golomb, 1970 ).

where ω < sub > 0 </ sub > is the undamped angular frequency of the oscillator and ζ is a constant called the damping ratio.

where Z ( s ) is ζ ( s ) multiplied by a gamma-factor, involving the gamma function.

where ζ is the damping ratio and ω < sub > 0 </ sub > is the natural frequency of the network.

where ζ is the damping ratio and ω < sub > 0 </ sub > is the natural frequency of the network.

where ζ lies on the unit circle and m < sub > i </ sub > is the multiplicity of the zero a < sub > i </ sub >, | a < sub > i </ sub >| < 1.

where n < sup > c </ sup > denotes the charge conjugate state, i. e., the antiparticle.

where we use the symbol k to denote the quantum numbers p and σ of the previous section and the sign of the energy, E ( k ), and a < sub > k </ sub > denotes the corresponding annihilation operators.

where r is the position vector of the particle relative to the origin, p is the linear momentum of the particle, and × denotes the cross product.

where and denotes the portfolio value at time t.

respectively, where e < sub > x </ sub >, e < sub > y </ sub >, e < sub > z </ sub > denotes the cartesian basis vectors ( all are orthogonal unit vectors ) and A < sub > x </ sub >, A < sub > y </ sub >, A < sub > z </ sub > are the corresponding coordinates, in the x, y, z directions.

where denotes the complex conjugate of.

denotes the rank-one operator that maps the ket to the ket ( where is a scalar multiplying the vector ).

Just as kets and bras can be transformed into each other ( making into ) the element from the dual space corresponding with is where A < sup >†</ sup > denotes the Hermitian conjugate ( or adjoint ) of the operator A.

where Z denotes J, Y, H < sup >( 1 )</ sup >, or H < sup >( 2 )</ sup >.

where C < sub > α </ sub > denotes I < sub > α </ sub > or e < sup > απi </ sup > K < sub > α </ sub >.

where ε denotes the Levi-Civita symbol, the metric tensor is used to lower the index on F, and the Einstein summation convention implies that repeated indices are summed over.

where denotes the length of a string p.

where J denotes the Jacobian matrix of partial derivatives of ƒ and is the induced norm on the matrix.

for every automorphism φ of G ( where φ ( H ) denotes the image of H under φ ).

where denotes the vector cross product and square brackets denote evaluation in the rotating frame of reference.

This functional relationship is often denoted y = f ( x ), where f denotes the function.

The common notation for the divergence ∇· F is a convenient mnemonic, where the dot denotes an operation reminiscent of the dot product: take the components of ∇ ( see del ), apply them to the components of F, and sum the results.

where the star denotes complex conjugation.

A *( B *( C * D )) where A, B, C, D are partially transparent image layers and "*" denotes a compositing operator ( with the left layer on top of the right layer ).

With this view it turns out that the extractor property is equivalent to: for any source of randomness that gives bits with min-entropy, the distribution is-close to, where denotes the uniform distribution on.

In mathematics and abstract algebra, a group is the algebraic structure, where is a non-empty set and denotes a binary operation called the group operation.

where denotes the largest integer that

where denotes the largest integer that is not greater than x.

where denotes Newton's gravitational constant,