Complementary Error Function Calculator
Calculate the complementary error function erfc(x) with step-by-step solutions and special values
erfc(x) = 1 - erf(x) = (2/√π) ∫ₓ^∞ exp(-t²) dt
Special values: erfc(0) = 1 | erfc(∞) = 0 | erfc(-x) = 2 - erfc(x)
erfc(x) ranges from 2 (at -∞) to 0 (at +∞) Valid range: 0 < y < 2
Quick Examples
Invalid Input
For erfcinv(y), the input must be between 0 and 2 (exclusive).
erfc() erfcinv()
erf() =
erf(x)
Q-function Q(x)
½ erfc(x)
Step-by-Step Solution
Identify the input value
x = y =
Apply the complementary error function formula Apply the inverse complementary error function
erfc(x) = 1 - erf(x) = (2/√π) ∫ₓ^∞ exp(-t²) dt Find x such that erfc(x) = y
Result
erfc() = erfcinv() =
Common Values Reference
| x | erfc(x) | erf(x) |
|---|---|---|
| -2 | 1.9953 | -0.9953 |
| -1 | 1.8427 | -0.8427 |
| 0 | 1.0000 | 0.0000 |
| 0.5 | 0.4795 | 0.5205 |
| 1 | 0.1573 | 0.8427 |
| 1.5 | 0.0339 | 0.9661 |
| 2 | 0.0047 | 0.9953 |
| 3 | 0.0000022 | 0.9999978 |
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About Complementary Error Function Calculator
What is the Complementary Error Function?
The complementary error function erfc(x) is a special function in mathematics closely related to the error function (erf). It is widely used in probability, statistics, and physics.
Mathematical Definition
The complementary error function is defined as:
erfc(x) = 1 - erf(x) = (2/√π) ∫ₓ^∞ exp(-t²) dt
Where erf(x) is the error function.
Relationship to Error Function
The complementary error function is simply:
erfc(x) = 1 - erf(x)
This relationship is particularly useful when erf(x) is close to 1, as erfc(x) provides greater numerical precision.
Key Properties
Special Values
- erfc(0) = 1
- erfc(∞) = 0
- erfc(-∞) = 2
Symmetry Property
erfc(-x) = 2 - erfc(x)
Asymptotic Behavior
For large x: erfc(x) ≈ exp(-x²)/(x√π)
Relationship to Q-Function
The Q-function used in digital communications is related to erfc:
Q(x) = ½ erfc(x/√2) erfc(x) = 2Q(x√2)
Relationship to Normal Distribution
For a standard normal distribution, the probability that a value exceeds x standard deviations:
P(X > x) = ½ erfc(x/√2)
Applications
- Probability and Statistics - Normal distribution calculations
- Signal Processing - Bit error rate calculations in communications
- Heat Transfer - Diffusion and conduction problems
- Physics - Particle diffusion, semiconductor physics
- Finance - Black-Scholes option pricing model
Common Values
| x | erfc(x) |
|---|---|
| 0 | 1.0000 |
| 0.5 | 0.4795 |
| 1.0 | 0.1573 |
| 1.5 | 0.0339 |
| 2.0 | 0.0047 |
| 2.5 | 0.0004 |
Inverse Complementary Error Function
The inverse function erfcinv(y) gives the value x such that erfc(x) = y:
erfcinv(erfc(x)) = x
Valid for y ∈ (0, 2).