function is: The probability density function is simply offset from the centered profile by : where: The mode and median are both located at . Derivatives Using the definition above for and , the first and second derivatives can be expressed in terms of the Faddeeva function as and respectively. Often, one or multiple Voigt profiles and/or their respective derivatives need to be fitted to a measured signal by means of Non-linear least squares, e.g., in spectroscopy. Then, further partial derivatives can be utilised to accelerate computations. Instead of approximating the Jacobian matrix with respect to the parameters , , and with the aid of finite differences, the corresponding analytical expressions can be applied. With and , these are given by: for the original voigt profile ; for the first order partial derivative ; and for the second order partial derivative . Since and play a relatively similar role in the calculation of , their respective partial derivatives also look quite similar in terms of their structure, although they result in totally different derivative profiles. Indeed, the partial derivatives with respect to and show more similarity since both are width parameters. All these derivatives involve only simple operations (multiplications and additions) because the computationally expensive and are readily obtained when computing . Such a reuse of previous calculations allows for a derivation at minimum costs. This is not the case for finite difference gradient approximation as it requires the evaluation of for each gradient respectively. Voigt functions The Voigt functions U, V, and H (sometimes called the line broadening function) are defined by where erfc is the complementary error function, and w(z) is the Faddeeva function. Relation to Voigt profile with and Numeric approximations Tepper-García Function The Tepper-García function, named after German-Mexican Astrophysicist Thor Tepper-García, is a combination of an exponential function and rational functions that approximates the line broadening function over a wide range of its parameters. It is obtained from a truncated power series expansion of the exact line broadening function. In its most computationally efficient form, the Tepper-García function can be expressed as where , , and . Thus the line broadening function can be viewed, to first order, as a pure Gaussian function plus a correction factor that depends linearly on the microscopic properties of the absorbing medium (encoded in ); however, as a result of the early truncation in the series expansion, the error in