Problem 37 - A First Course in Complex Analysis with Applications

Question

In Problems 37 and 38 , 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

\( e^{(1+i)/10} \approx 1.1 + 0.11i \) using three terms.

Step 1: Recall the Maclaurin Series

The Maclaurin series for a function \( f(x) \) is given by: \[ f(x) = f(0) + f'(0)x + \frac{f''(0)x^2}{2!} + \frac{f'''(0)x^3}{3!} + \ldots \]For the exponential function \( e^x \), the series becomes: \[ e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \ldots \]

Step 2: Identify Terms Needed

We need to approximate \( e^{(1+i)/10} \) using the first three terms of the Maclaurin series for \( e^x \). This means:\[ e^{(1+i)/10} \approx 1 + \frac{1+i}{10} + \frac{\left(\frac{1+i}{10}\right)^2}{2!} \]

Step 3: Calculate the First Term

The first term is simply \( 1 \), from the constant term in the series.

Step 4: Calculate the Second Term

Calculate the second term by substituting \( x = \frac{1+i}{10} \) into \( x \): \[ \frac{1+i}{10} = 0.1 + 0.1i \] So the second term is \( 0.1 + 0.1i \).

Step 5: Calculate the Third Term

The third term is obtained by using: \[ \frac{x^2}{2!} = \frac{\left(\frac{1+i}{10}\right)^2}{2} \]Calculate \( \left(\frac{1+i}{10}\right)^2 \): \[ \left(\frac{1+i}{10}\right)^2 = \left(0.1 + 0.1i\right)^2 = 0.01 + 0.02i + (0.1i)^2 = 0.01 + 0.02i - 0.01 = 0.02i \] Now divide by 2: \[ \frac{0.02i}{2} = 0.01i \] So the third term is \( 0.01i \).

Step 6: Sum All Terms

Add together all the terms to approximate \( e^{(1+i)/10} \):\[ e^{(1+i)/10} \approx 1 + 0.1 + 0.1i + 0.01i = 1.1 + 0.11i \]

Key Concepts

Complex Numbers

Complex numbers are a fascinating extension of our familiar number system. Where traditional numbers can be placed on a line, complex numbers have a real part and an imaginary part. This means each complex number can be represented as a point on a plane.

• **Real part**: This is equivalent to the numbers we deal with every day, for example, in the expression \(1+i\), 1 is the real part.
• **Imaginary part**: The term that involves the square root of -1, denoted as \(i\). In \(1+i\), \(i\) is the imaginary part.

Complex numbers are often written in the form \(a + bi\), where \(a\) represents the real part and \(b\) represents the coefficient of the imaginary part.
These numbers have special properties that allow them to be added, subtracted, multiplied, and divided just like real numbers, but with the extra rule that \(i^2 = -1\). Complex numbers are crucial when dealing with any kind of waveforms in engineering and physics, allowing for elegant mathematical solutions to complicated problems.

Exponential Function

The exponential function is a mathematical expression popular for modeling growth and decay processes. Defined as \(e^x\) where \(e\) is approximately 2.71828, it holds a special place in mathematics due to its unique properties.

• **Derivative and Integral**: The derivative and integral of \(e^x\) are both \(e^x\), making calculations simpler in calculus.
• **Growth**: It describes how quantities grow continuously and exponentially, an essential concept in fields like biology and finance.

This makes the exponential function particularly interesting in both real and imaginary realms. For complex numbers, it creates beautiful patterns like spirals on an Argand diagram. When used with complex numbers, especially in Euler's formula \(e^{ix} = \cos{x} + i\sin{x}\), it bridges exponential functions with trigonometric functions in a fascinating way.

Series Approximation

Series approximation is a powerful mathematical technique to estimate values of functions using polynomials. The Maclaurin series, a type of Taylor series centered at zero, is particularly useful for approximating functions like the exponential function.

• **Basics of Maclaurin Series**: It expresses a given function as an infinite sum of terms calculated from the derivatives of the function. For example, for \(e^x\), the Maclaurin series is \(1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \ldots\).
• **Application**: By truncating the series to a finite number of terms, complex functions like \(e^{(1+i)/10}\) can be approximated simply by calculating a few derivatives and constants.
• **Use in Calculations**: This is extremely helpful when calculating values that are difficult to evaluate exactly, as seen in computer science and numerical analysis.

Series approximation balances accuracy with simplicity and is widely used in scientific calculations and engineering models where exact answers are not feasible, offering close enough values for practical purposes.