Engineering & Scientific
Gaussian Error Function & Complementary Tail (ERF & ERFC) for Fickian Diffusion & BER
Evaluate erf(x) and erfc(x) = 1 - erf(x) to 6-decimal accuracy for semi-infinite Fick's Second Law dopant/heat diffusion profiles and BPSK digital communication bit-error rates.
Where ERF and ERFC govern physical transport and signal noise
The Gaussian error function erf(x) = (2 / sqrt(pi)) * integral_0^x exp(-u^2) du and its complement erfc(x) = 1 - erf(x) solve the one-dimensional diffusion equation under constant surface concentration: C(z, t) / C_surface = erfc(z / (2 * sqrt(D * t))).
Evaluating erfc(|x|) directly via Abramowitz and Stegun's 5-term rational factor multiplied by EXP(-x^2) (7.1.26, error <= 1.5e-7) preserves relative tail accuracy even at large penetration depths or high Eb/N0 signal-to-noise ratios.
| Similarity Variable eta | erf(eta) | erfc(eta) = C / C_s | Physical Meaning |
|---|---|---|---|
| 0.00 | 0.0000 | 1.0000 | Boundary surface concentration (z = 0) |
| 0.50 | 0.5205 | 0.4795 | Near-surface diffusion zone |
| 1.00 | 0.8427 | 0.1573 | Characteristic diffusion length z = 2*sqrt(D*t) |
| 2.00 | 0.9953 | 0.0047 | Deep bulk tail (< 0.5% of surface level) |
Dimensionless Diffusion Similarity Argument eta
Divide depth z (meters) by the characteristic diffusion scale 2 * SQRT(Diffusivity * Time).
=B2 / (2 * SQRT(B3 * B4))Direct ERFC & ERF Rational Evaluation
Compute erfc(eta) directly so C(z, t) = C_surface * erfc(eta), and recover erf(eta) = 1 - erfc(eta).
=LET(eta, B6, x_abs, ABS(eta), poly_t, 1 / (1 + 0.3275911 * x_abs), erfc_abs, poly_t * (0.254829592 + poly_t * (-0.284496736 + poly_t * (1.421413741 + poly_t * (-1.453152027 + 1.061405429 * poly_t)))) * EXP(-(x_abs ^ 2)), IF(eta = 0, 1, ROUND(IF(eta > 0, erfc_abs, 2 - erfc_abs), 4)))Try the example
Use the original values or change them to test your understanding.