Question

Find the power series for \(e^{x} \cos x\) and for \(e^{x} \sin x\) from the series for \(e^{z}\) in the following way: Write the series for \(e^{z} ;\) put \(z=x+\) iy. Show that \(e^{z}=e^{x}(\cos y+i \sin y) ;\) take real and imaginary parts of the equation, and put \(y=x.\)

Short answer

The power series for \( e^{x} \text{cos}(x) \) and \( e^{x} \text{sin}(x) \) are derived by writing \( e^z \) with \( z = x + iy \), then separating real and imaginary parts, finally letting \( y = x \).

Step-by-step solution

Write the series for \( e^z \)

The power series expansion for \( e^z \) is given by \[ e^z = \frac{z^0}{0!} + \frac{z^1}{1!} + \frac{z^2}{2!} + \frac{z^3}{3!} + \frac{z^4}{4!} + \frac{z^5}{5!} + \frac{z^6}{6!} + \frac{z^7}{7!} + \frac{z^8}{8!} + \frac{z^9}{9!} + \frac{z^{10}}{10!} + \frac{z^{11}}{11!} + \frac{z^{12}}{12!} + \frac{z^{13}}{13!} + \frac{z^{14}}{14!} + \frac{z^{15}}{15!} + \frac{z^{16}}{16!} + \frac{z^{17}}{17!} + \frac{z^{18}}{18!} + \frac{z^{19}}{19!} + \frac{z^{20}}{20!} \]

Substitute \( z = x + iy \)

Substitute \( z = x + i y \) in the series:\[ e^{x + i y} = \frac{(x + i y)^0}{0!} + \frac{(x + i y)^1}{1!} + \frac{(x + i y)^2}{2!} + \frac{(x + i y)^3}{3!} + \frac{(x + i y)^4}{4!} + \frac{(x + i y)^5}{5!} + \frac{(x + i y)^6}{6!} + \frac{(x + i y)^7}{7!} + \frac{(x + i y)^8}{8!} + \frac{(x + i y)^9}{9!} + \frac{(x + i y)^{10}}{10!} \]

Expand using Euler's Formula

Using Euler's formula \( e^{i y} = \text{cos}(y) + i \text{sin}(y) \), the series can be rearranged:\[ e^{x + i y} = e^x \big(\text{cos}(y) + i \text{sin}(y)\big) \]

Separate Real and Imaginary Parts

Separate the expanded form into its real and imaginary parts:\[ e^{x + i y} = e^x \text{cos}(y) + i e^x \text{sin}(y) \]The real part is \( e^x \text{cos}(y) \) and the imaginary part is \( e^x \text{sin}(y) \).

Substitute \( y = x \)

Finally, let \( y = x \) to find the desired series:\[ e^{x} \text{cos}(x) \text{ and } e^{x} \text{sin}(x) \]

Key concepts

Euler's Formula

Euler's formula is a key concept in complex analysis, connecting complex exponentials with trigonometric functions. It states that for any real number y, the equation \(e^{iy} = \cos(y) + i\sin(y)\) holds. This connection is powerful because it allows us to express complex exponentials in terms of real and imaginary parts. Euler's formula is fundamental when working with power series involving complex numbers, as it simplifies the manipulation of expressions involving imaginary units.
By applying Euler's formula, we can decompose complex functions into more manageable trigonometric components, which is particularly useful in fields like electrical engineering, physics, and signal processing.

Complex Analysis

Complex analysis is the study of functions that operate on complex numbers. In this exercise, we deal with the function \(e^z\), where \(z\) is a complex variable written as \(x + iy\). This field of mathematics provides tools to handle and manipulate complex functions, making it possible to extend real analysis concepts into the complex plane.
One significant aspect of complex analysis is the power series expansion, which we use to break down functions into infinite sums of terms. For example, the power series for \(e^z\) is given by:
\[ e^z = \sum_{n=0}^{\infty} \frac{z^n}{n!} \]
Substituting \(z = x + iy\) in this series lets us apply Euler's formula and separate the expression into real and imaginary parts.
Understanding these concepts allows us to solve complex problems more efficiently, using powerful techniques that exploit the properties of complex numbers and their functions.

Real and Imaginary Parts

In complex numbers, each number has a real part and an imaginary part. For a complex number \(z = x + iy\), \(x\) is the real part, and \(iy\) is the imaginary part. Working with both parts separately can simplify complex expressions and problem-solving.
When we substitute \(z = x + iy\) into the power series expansion of \(e^z\), we can use Euler's formula to obtain:
\[ e^z = e^x (\cos(y) + i \sin(y)) \]
Here, \( e^x \cos(y) \) becomes the real part, and \( e^x \sin(y) \) becomes the imaginary part. This separation is helpful in many applications, including solving differential equations and evaluating integrals.
Finally, by setting \(y = x\), we specifically find expressions for \(e^{x} \cos(x)\) and \(e^{x} \sin(x)\), making the complex expression more intuitive and easier to handle.