Problem 36 - A First Course in Complex Analysis with Applications

Question

The error function \(\operatorname{erf}(z)\) is defined by the integral \(\operatorname{erf}(z)=\frac{2}{\sqrt{\pi}} \int_{0}^{z} e^{-t^{2}} d t\) Find a Maclaurin series for erf \((z)\) by integrating the Maclaurin series for \(e^{-t^{2}}\).

Step-by-Step Solution

Answer

The Maclaurin series for \(\operatorname{erf}(z)\) is \(\frac{2}{\sqrt{\pi}}(z - \frac{z^3}{3} + \frac{z^5}{10} - \cdots)\).

Step 1: Find Maclaurin Series for the Function

Start with the exponential function series: \ e^{-t^2} = 1 - t^2 + \frac{t^4}{2!} - \frac{t^6}{3!} + \cdots \ This is the Maclaurin series for \( e^{-t^2} \).

Step 2: Integrate the Series Term by Term

Integrate each term of the Maclaurin series for \( e^{-t^2} \) from 0 to \( z \): \ \int_{0}^{z} e^{-t^2} dt = \int_{0}^{z} \left( 1 - t^2 + \frac{t^4}{2!} - \frac{t^6}{3!} + \cdots \right) dt \ = \left[ t - \frac{t^3}{3} + \frac{t^5}{5 \cdot 2!} - \frac{t^7}{7 \cdot 3!} + \cdots \right]_{0}^{z} \ = z - \frac{z^3}{3} + \frac{z^5}{10} - \frac{z^7}{42} + \cdots \

Step 3: Multiply by the Constant Factor

The original definition for \( \operatorname{erf}(z) \) includes the factor \( \frac{2}{\sqrt{\pi}} \). Multiply the integrated series by this factor to find \( \operatorname{erf}(z) \): \ \operatorname{erf}(z) = \frac{2}{\sqrt{\pi}} \left( z - \frac{z^3}{3} + \frac{z^5}{10} - \frac{z^7}{42} + \cdots \right). \ This is the Maclaurin series for \( \operatorname{erf}(z) \).

Key Concepts

Error Function

The error function, denoted as \( \operatorname{erf}(z) \), is a special function that plays a crucial role in probability, statistics, and partial differential equations. It is defined through an integral:

• \( \operatorname{erf}(z) = \frac{2}{\sqrt{\pi}} \int_{0}^{z} e^{-t^{2}} \, dt \)

This function essentially measures the probability that a random variable following a normal distribution deviates from its mean by more than a specified value. It's heavily used in error analysis and heat conduction problems.
The integration limits, from \(0\) to \(z\), reflect how this function gives cumulative distribution values up to \(z\). The factor \(\frac{2}{\sqrt{\pi}}\) scales this integral to normalize the function, making it crucial for comparative analysis in standard normal distributions.

Integration of Series

When dealing with functions that have a known Maclaurin series, such as \( e^{-t^2} \), you can find the integral of the entire series by integrating term by term. This makes it easier to find functions like \( \operatorname{erf}(z) \) which are defined integrally. For \( e^{-t^2} \), its Maclaurin series is:

• \( e^{-t^2} = 1 - t^2 + \frac{t^4}{2!} - \frac{t^6}{3!} + \cdots \)

By integrating each term separately from \(0\) to \(z\), we get:

• \( \int_{0}^{z} e^{-t^2} \, dt = z - \frac{z^3}{3} + \frac{z^5}{10} - \frac{z^7}{42} + \cdots \)

This step-by-step integration allows you to construct a series expansion for integral-based functions, facilitating their computation for small values of \(z\). This approach is not only precise but also manageable, especially when dealing with infinite sums.

Exponential Function Series

The exponential function series is foundational in calculus and analysis, forming the basis for expressing many functions as infinite sums. In the case of \( e^{-t^2} \), the series aids in simplifying the process of finding \( \operatorname{erf}(z) \). The general form of the exponential function series is derived from \( e^x \) and adapted for specific functions:

• \( e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots \)
• For \( e^{-t^2} \), modify as: \( e^{-t^2} = 1 - t^2 + \frac{t^4}{2!} - \frac{t^6}{3!} + \cdots \)

This manipulation of the exponential function series allows complex functions to be broken down step by step. This process is especially useful for functions that are difficult to integrate directly, like in the case of the error function, providing a clear path to approximate values through series.

Complex Analysis

Complex analysis, although not the primary focus of developing the series for \( \operatorname{erf}(z) \), provides essential tools and understanding for such functions. Functions like \( \operatorname{erf}(z) \) arise naturally within the scope of complex analysis, especially when dealing with integrals of complex functions and analytic continuations.
By extending real-valued functions into the complex plane, mathematicians can explore identities and relationships that aren't apparent in real analysis alone. For the error function:

• Complex analysis enables extensions to complex variables, offering insights into their symmetry and convergence properties.
• This broadens the applications of \( \operatorname{erf}(z) \) beyond standard problems, to include diverse fields like quantum physics and electrical engineering.

Leveraging complex analysis enriches our understanding and expands the utility of integral functions like the error function, providing deeper insights into their behavior and applications.