Problem 33 - Advanced Engineering Mathematics

Question

In Problems 33 and 34, approximate the value of the given expression using the indicated number of terms of a Maclaurin series. $$ e^{(1+i) / 10} \text {, three terms } $$

Step-by-Step Solution

Answer

The approximation is \(1.1 + 0.11i\).

Step 1: Understand the Maclaurin Series

A Maclaurin series is a Taylor series expanded at zero. For a function \( f(x) \), the Maclaurin series is \( f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \ldots \). Our goal is to approximate \( e^{(1+i)/10} \) using three terms from its Maclaurin series.

Step 2: Identify the Function and Expand

We are dealing with \( f(x) = e^x \). The Maclaurin expansion of \( e^x \) is \( 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \ldots \). Substitute \( x = \frac{1+i}{10} \) into this series. We will use the first three terms: \(1 + \frac{1+i}{10} + \frac{\left(\frac{1+i}{10}\right)^2}{2} \).

Step 3: Substitute the Value

Substitute \( x = \frac{1+i}{10} \) into the first three terms of the series: \( 1 + \frac{1+i}{10} + \frac{\left(\frac{1+i}{10}\right)^2}{2} \). This leads to the expression: \( 1 + \frac{1}{10} + \frac{i}{10} + \frac{(1+i)^2}{200} \).

Step 4: Compute Each Term

Calculate each term: \(\frac{1}{10} = 0.1\), \(\frac{i}{10}\), and \((1+i)^2 = 1 + 2i - 1 = 2i\). Then, \(\frac{2i}{200} = \frac{i}{100} = 0.01i\). Thus, the expression simplifies to \( 1 + 0.1 + \frac{i}{10} + 0.01i \).

Step 5: Simplify the Expression

Combine the real and imaginary parts: \((1 + 0.1) + (\frac{i}{10} + 0.01i)\) which results in \(1.1 + 0.11i\).

Step 6: Final Approximation

Therefore, the approximation of \( e^{(1+i)/10} \) using three terms of the Maclaurin series is \(1.1 + 0.11i\).

Key Concepts

Taylor series

The Taylor series is a powerful mathematical concept used to approximate functions with polynomials. It does this by expanding a function around a specific point. This point can be any number, but when it is zero, the Taylor series becomes a Maclaurin series.

The general formula for a Taylor series of a function \( f(x) \) around the point \( a \) is:

• \( f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \ldots \)

In simple terms, it means breaking down the function into an infinite sum of terms calculated from the values of its derivatives at a single point.

For the Maclaurin series, which is a special case of the Taylor series, \( a = 0 \). This simplifies the expansion to:

• \( f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \ldots \)

Understanding this is crucial because it allows us to approximate complex functions like exponentials, logarithms, and trigonometric functions with easier-to-use polynomials.

Complex exponential function

The complex exponential function is an extension of the regular exponential function \( e^x \) to complex numbers. In mathematics and engineering, it has countless applications including solving differential equations and modeling wavefunctions in quantum mechanics.

For a complex number \( z = x + yi \), the exponential function is defined as:

• \( e^z = e^{x+yi} = e^x (\cos(y) + i\sin(y)) \)

This is based on Euler's formula, a fundamental bridge connecting analysis and trigonometry.

When we apply this concept to our problem, we might need to express the complex exponential in terms of its components. The expression \( e^{(1+i)/10} \) is handled like any complex number, breaking it down into real and imaginary parts with the help of the Maclaurin series.

By expanding the exponential function using the Maclaurin series, we transform this problem into a manageable series of simpler calculations involving basic arithmetic and trigonometry.

Series approximation

Series approximation involves using partial sums of a series to approximate complex functions. This is particularly useful when an exact solution is not possible or necessary.

By using only the first few terms of the Maclaurin or Taylor series, we create a polynomial that closely resembles the original function over a certain range.

In our exercise, we're using just three terms to approximate \( e^{(1+i)/10} \). Each added term in a series expansion moves our approximation closer to the true value. However, computation beyond a certain point may yield only marginal accuracy improvements, especially in cases where \( x \) is small and the higher powers become negligible.

The steps include substituting the variable in the series with the given number, calculating each part, and then combining the results. Remember:

• Accuracy improves with more terms but at a cost of increased complexity.
• The method provides insight into the behavior of the function near the expansion point.

This makes series approximation a robust tool for hypothesizing outcomes and solving real-world tasks where precision and simplicity are needed in tandem.