Global approximation for some functions
Approximate closed form representations of functions are useful for mathematical manipulations. Nonlinear sequence transforms can be used to evaluate the function using a few terms of the series representation of the function and these transforms can be used for functions with complex argument as well. Moreover, if an asymptotic expansion of the function is available, an approximant for the function, valid for the entire range of the variable, can be obtained with Pad approximants as well as Levin and Weniger transforms. In addition, one can obtain an approximation for a function using quadratic Pad approximation which is also valid for the entire range of the variable. We demonstrate this for some functions frequently encountered in scientific problems. These include the error function, the Fresnel integral, the Dawson integral, the Euler integral and the elliptic integral. A comparison is made between the approximants obtained with Pad approximants and those generated by Levin and Weniger transforms.
asymptotic expansion quadratic pad approximant fresnel integrals dawson integral elliptic integrals multi-point levin and weniger tansforms error integral