Problem 33 - Advanced Engineering Mathematics

Question

Approximate the value of the given expression using the indicated number of terms of a Maclaurin series. . \(e^{(1+i) 10}\), three terms \(\quad\)

Step-by-Step Solution

Answer

The approximation is \(11 + 110i\).

Step 1: Recall the Maclaurin Series for \(e^x\)

The Maclaurin series for \(e^x\) is given by the expression \(1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots\). This series provides the approximation of \(e^x\) using as many terms as needed for the desired accuracy.

Step 2: Apply the Maclaurin Series to \(e^{(1+i)10}\)

We will substitute \((1+i)10\) into the Maclaurin series \(e^x\). For this problem, we only need the first three terms. Therefore, our expression becomes: \(1 + (1+i)10 + \frac{((1+i)10)^2}{2!}\).

Step 3: Calculate Each Term

Start calculating each of the first three terms:- The zeroth term is simply \(1\).- The first term is \((1+i)10 = 10 + 10i\).- To find the second term, calculate \(((1+i)10)^2 = (10+10i)^2 = 100 + 200i + 100i^2 = 100 + 200i - 100 = 200i\), since \(i^2 = -1\). Then, divide by \(2!\) to get \(\frac{200i}{2} = 100i\).

Step 4: Combine the Terms

Add the results from each term: - The zeroth term contributes \(1\).- The first term contributes \(10 + 10i\).- The second term contributes \(100i\). Thus, combining these gives: \(1 + 10 + 10i + 100i = 11 + 110i\).

Key Concepts

Exponential Functions

Exponential functions are a crucial part of mathematics, frequently used to model growth and decay in various fields such as finance, biology, and physics. The exponential function is characterized by its base, "e," which is approximately equal to 2.71828. This number is unique in that the function \( e^x \) has the same rate of growth at any point as the value of the function itself. This means that the derivative of \( e^x \) is also \( e^x \), creating a consistent pattern.

Exponential functions model phenomena where the rate of change is proportional to the current value, like population growth. In the context of complex numbers, they are extended to use imaginary exponents, written as \( e^{ix} \), which links to Euler's formula: \( e^{ix} = \cos(x) + i\sin(x) \). This formula creates a powerful connection between exponential functions and trigonometry, enriching the understanding and application of exponential functions in complex analysis.

Understanding \( e^{x} \) through its series expansion, like the Maclaurin Series, gives an insight into how functions can be approximated with polynomials for practical computation.

Complex Numbers

Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using two components: a real part and an imaginary part. A complex number is usually written as \( a + bi \), where \( a \) is the real part, \( b \) is the imaginary part, and \( i \) is the imaginary unit, defined by \( i^2 = -1 \). This means one can represent complex numbers as points or vectors in a plane, providing great utility in engineering and physics.

When dealing with the Maclaurin series for expressions like \( e^{(1+i)x} \), complex numbers come into play by allowing manipulations involving both real and imaginary components. The real component affects the function's magnitude, while the imaginary component influences the phase. The power of complex numbers lies in their ability to elegantly express phenomena that change cyclically, such as waves and oscillations, which can be vital in analyzing alternating current circuits."

Working with complex numbers in series approximations involves squaring and combining terms, as seen in the exercise where \((10+10i)^2\) demonstrates the impact of \(i^2 = -1\), converting oscillatory behavior into real contributions.

Series Approximation

Series approximation is a mathematical technique that allows complex functions to be expressed as a sum of simpler terms. In cases where a precise analytical solution is difficult or impossible to obtain, using a series can provide a convenient and powerful way to approximate function values over an interval.

The Maclaurin series is a specific type of series approximation used for functions differentiable at zero. It expresses a function as a sum of its derivatives at zero, allowing the function \( f(x) \) to be approximated as \( f(0) + f'(0)x + \frac{f''(0)x^2}{2!} + \cdots \). This is particularly useful for complicated or unfamiliar functions like exponential functions with complex exponents.

Using only a few terms of this series can offer a sufficiently accurate approximation for many practical purposes, saving time and resources in computation. By approximating \( e^{(1+i)10} \) with just the first three terms, as the exercise demonstrated, we effectively calculate a close approximation using manageable arithmetic. Series approximations make tasks feasible when dealing with complex math, simplifying the procedures down to basic calculations.