Properties of Error Function

While the erf (a ) is itself an integral function, nonetheless it can also be integrated again, thus the indefinite integral 

For continuous computation, the erf (x) can be given an asymptotic expansion (Abramowitz and Stegun 1965) or it can be represented in terms of the Confluent Hypergeometric function after Kummer (e.g., see Example 3.4). A rational approximation given by C. Hastings in Abramowitz and Stegun (1965) is useful for digital computation 

This approximation ensures erf( ) = 1. Occasionally, the complementary error function is convenient it is defined as 

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