Question

Evaluate the following, using either power series, a table of error functions, or asymptotic series, whichever is appropriate. $$ 1-\operatorname{erf}(5)=\operatorname{erfc}(5) $$

Short answer

The value of \(\text{erfc}(5) \) is approximately \(1.5374598 \times 10^{-12}\).

Step-by-step solution

Identify the Expression

The problem requires evaluating an error function complement at a specific point: \(\text{erfc}(5)\).

Use the Complementary Error Function Table

Look up the value of the complementary error function \(\text{erfc}(x)\) from a standard table. Specifically, note the value for \(x = 5\).

Consult the Values from the Table

From the table of complementary error functions, the value of \(\text{erfc}(5)\) is very close to \(1.5374598 \times 10^{-12}\).

Key concepts

erfc function

The complementary error function, denoted as \(\text{erfc}(x)\), is a mathematical function closely related to the error function \(\text{erf}(x)\). The relationship between them is given by the equation: \[ \text{erfc}(x) = 1 - \text{erf}(x) \]. This function is particularly useful in probability, statistics, and partial differential equations. For large values of \ 'x' \, \(\text{erfc}(x)\) allows for calculating the tail probabilities of the normal distribution.

Using tables of the complementary error function can provide accurate values for specific points, which are often used in solving integrals and differential equations. Many scientific software libraries also include efficient algorithms for computing \(\text{erfc}(x)\).

A key aspect of \(\text{erfc}(x)\) is its rapid decay to zero as \ 'x' \ becomes large. This rapid decay is leveraged in applications where high precision is necessary.

asymptotic series

An asymptotic series provides an approximate representation of functions, especially useful for large values of variables. The complementary error function \(\text{erfc}(x)\) can be approximated by an asymptotic expansion for large \'x'\. This series allows for calculations when exact values are hard to compute.

For \(\text{erfc}(x)\) as \ 'x' \ becomes large:

\[\text{erfc}(x) \approx \frac{e^{-x^2}}{x \sqrt{\pi}} \left( 1 - \frac{1}{2x^2} + \frac{3}{4x^4} - \cdots \right)\]
This series converges more rapidly for larger values of \ 'x' \. Practically, only a few terms are needed to get a good approximation. Use of asymptotic series thus balances computational efficiency with desired accuracy and is highly valued in analytical and numerical mathematics.

error functions table

Tables of error functions, including the complementary error function \(\text{erfc}(x)\), are useful tools for quickly finding values of these functions. These tables provide pre-calculated values for given points, which saves time compared to numerical computation.

For the given problem, the value of \(\text{erfc}(5)\) from these tables is approximately \ 1.5374598 \times 10^{-12}\. Such tables are typically found in statistical and mathematical reference books. They can also be accessed electronically through online databases and software libraries.

Using these tables ensures accuracy and is essential for verifying results obtained through other methods like asymptotic series or power series expansions.