The Complementary Error Function

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The Complementary Error FunctionFrank R. KschischangDepartment of Electrical & Computer EngineeringUniversity of TorontoApril 10, 2017The Complementary Error Function , erfc(x), is defined, forx 0, aserfc(x) = 2 x1 exp( u2) Complementary Error Function represents the area under the two tails of a zero-meanGaussian probability density Function with variance 2= 1/2, as illustrated in Fig. 1. Theso-called Error Function , erf(x), is defined viaerf(x) = 1 erfc(x).From the fact that a probability density Function has unit integral, we see thaterfc(0) = Complementary Error Function erfc(x) is plotted in Fig.

0 0:5 1 1:5 2 2:5 3 3:5 4 10 8 10 7 10 6 10 5 10 4 10 3 10 2 10 1 100 exp( 2x ) x p ˇ 2exp( 2x ) p ˇ(x+ x2+2) x erfc(x Bounds Upper bound Lower bound erfc(x) Figure 2: The function erfc(x) plotted together with an upper bound and a lower bound as