The Erfc function is initially defined in the math.h header file and serves the purpose of computing the complementary error function for a specified input. This function holds significance in probability theory, statistics, as well as various fields of engineering and science.
Syntax:
It has the following syntax:
#include <math.h>
double erfc(double x);
Parameter:
X: This is a data type that stores a double value for inputting into the complementary error function. It can accept any numerical value.
Return Value:
The erfc function provides a numerical output of type double, representing the calculated complementary error function for the given input x.
| Parameter | Return Value |
|---|---|
| x = + ∞ | +0 |
| x= - ∞ | 2 |
| x= nan | nan |
Complementary error function in mathematics
In the integral provided, the input to the erfc function functions as the starting point, and x must be smaller than this threshold.
The error function is a commonly employed concept in the fields of probability and statistics. It involves the calculation of the area under the normal distribution curve, providing the probability that a random variable Y, which follows a normal distribution with a mean of 0 and a variance of ½, lies within the interval [−x, x].
The complementary error function and the series representation provide approximations for small and large values of x, respectively.
The diagram displayed below illustrates the graph of the complementary error function (erfc):
The diagram provided below demonstrates the functionality of the erfc function:
Example 1:
Below is an illustration of a C++ program that computes the complementary error function.
#include <stdio.h>
#include <math.h> // Include the math library for mathematical functions
int main()
{
// Define a float variable y and assign it a value of positive infinity
float y = 2.0 / 0.0;
// Output the value of y
printf("Value of y is: %f\n", y);
// Calculate and output the complementary error function of y
printf("erfc(y): %f\n", erfc(y));
return 0; // Return 0 to indicate successful completion of the program
}
Output:
Value of y is: inf
erfc(y): 0.000000
=== Code Execution Successful ===
Example 2:
#include <stdio.h>
#include <math.h> // Include the math library for mathematical functions
int main()
{
// Define a float variable z and assign it a value of negative infinity
float z = -1.0 / 0.0;
// Output the value of z
printf("Value of z is: %f\n", z);
// Calculate and output the complementary error function of z
printf("erfc(z): %f\n", erfc(z));
return 0; // Return 0 to indicate successful completion of the program
}
Output:
Value of z is: -inf
erfc(z): 2.000000
=== Code Execution Successful ===
Example 3:
#include <stdio.h>
#include <math.h> // Include the math library for mathematical functions
int main()
{
// Define a float variable w and assign it a value of NaN
float w = sqrt(-2); // sqrt of a negative number results in NaN
// Output the value of w
printf("Value of w is: %f\n", w);
// Calculate and output the complementary error function of w
printf("erfc(w): %f\n", erfc(w));
return 0; // Return 0 to indicate successful completion of the program
}
Output:
Value of w is: -nan
erfc(w): -nan
=== Code Execution Successful ===
Application of erfc
The complementary error function erfc plays a vital role in a wide range of disciplines such as statistics, engineering, physics, and finance. Below are several important use cases:
1. Statistics and Probability
- Normal Distribution: Actually, erfc is the complement of the cumulative distribution function (CDF) of the standard normal distribution. It is used to find the probability/percentile of a normally distributed variable in a population.
- Hypothesis Testing: In statistical hypothesis testing, erfc is used in the computation of the critical values and the p-values in hypothesis testing that involves the normal distribution.
- Diffusion Processes: The erfc is used to solve diffusion phenomena, such as in the heat equation. It helps define the variation of particles or heat at any given time. where C(x,t) is the concentration, C_0 is the initial concentration, D is the diffusion coefficient, and t is time.
- Signal Processing: The erfc is utilized in the error analysis applications for communication systems where the probability of error in the digital communication systems is to be determined.
- Quantum Mechanics: In quantum mechanics, the erfc is involved in the solutions of Schrödinger's equation for specific forms of the potentials.
- Error Analysis: Statistically, physicists adopt it as the complementary of the error function to analyze the errors in measurement and experiments.
2. Engineering
3. Physics
Option Pricing: Within financial mathematics, specifically in the Black-Scholes model, erfc serves the purpose of calculating the probabilities linked to the normal distribution of stock prices.
5. Other Scientific Fields
- Biology and Medicine: In pharmacokinetics erfc is applied to express the distribution and elimination of the drugs in the body at intervals of time.
- Environmental Science: It is applied mainly in simulating the concentration distribution of pollutants in the air or water.
- Integral Computation: When dealing with the erfc function, it is commonly employed to compute integrals involving Gaussian functions, which are prevalent in various fields of science and engineering.
- Heat Transfer: In heat transfer problems, the complementary error function is used to describe the temperature distribution in a semi-infinite solid subjected to a sudden change in surface temperature. where T(x,t) is the temperature at position x and time t, T 0 is the initial temperature, and α is the thermal diffusivity.
- Radiation Therapy: In radiation therapy for cancer treatment, erfc can be used to model the dose distribution of radiation delivered to tissues.
- Astrophysics: Used in the analysis of light curves and other data from celestial objects to model the distribution of errors and uncertainties.
6. Numerical Computation
Examples of Specific Applications
The flexibility and extensive array of uses of the erfc function establish it as a crucial asset in various scientific and engineering fields.
Conclusion:
In summary, the erfc function, which is the counterpart of the error function, is a significant mathematical tool within the math.h library. This header file is integral to the C programming language and finds application across a range of disciplines including statistics, engineering, physics, and finance due to its unique capabilities and applications.
For this purpose, the complementary error function (erfc) holds importance in statistical and probabilistic contexts, particularly in normal distribution applications for probability assessment and significance test analysis in hypothesis testing. In engineering applications, erfc is commonly utilized in resolving diffusion challenges and contributes significantly to error assessment in communication systems. In the field of quantum mechanics, erfc emerges in error propagation equations studied by physicists and is a component of option pricing models like the Black-Scholes model favored by financial experts.
This function is not limited to specific fields; it is utilized in biology and medicine to simulate drug distribution, in environmental science to explain pollutant dispersion, and in engineering for heat transfer issues. Furthermore, it can assist in solving integrals involving Gaussian functions frequently encountered in scientific and engineering literature.