theta1

Theta1, written as theta1 or θ1, is a symbolic label used across mathematics, science, and engineering to denote the first component of a set of quantities named theta. Because theta is a common symbol for angles and parameters, theta1 is typically the first angle, the first parameter in a vector θ = (θ1, θ2, ...), or the first member of a sequence.

In mathematics, theta1 most often refers to the Jacobi theta function θ1(z|τ). This is one of the four Jacobi theta functions, an odd, quasi-periodic function of a complex variable z with a complex parameter τ in the upper half-plane. It can be defined by a Fourier series, for example θ1(z|τ) = 2 ∑_{n=0}^∞ (-1)^n q^{(n+1/2)^2} sin((2n+1)z), where q = e^{iπτ}. It has zeros at z = mπ + nπτ for integers m and n, and satisfies notable transformation properties under shifts by π and πτ.

Beyond the function, theta1 is used as a general label for the first angle in a two-angle description in geometry, or as the first coordinate of a parameter vector θ in optimization and statistics. In many texts context clarifies its meaning. In some domains, theta1 may denote a specific variable or function unrelated to the Jacobi theta function; without context, the meaning of theta1 remains ambiguous.