Question

Define \(\sin z\) and \(\cos z\) by their power series. Write the power series for \(e^{i z} .\) By comparing these series obtain the definition (11.4) of \(\sin z\) and \(\cos z.\)

Short answer

Using power series definitions for \sin z and \cos z, the equation \ e^{i z} = \cos z + i \sin z \ is derived by comparing the series.

Step-by-step solution

Define \sin z by its power series

The power series definition for \sin z is: \[ \sin z = \sum_{n=0}^{\infty} (-1)^n \frac{z^{2n+1}}{(2n+1)!} \]

Define \cos z by its power series

The power series definition for \cos z is: \[ \cos z = \sum_{n=0}^{\infty} (-1)^n \frac{z^{2n}}{(2n)!} \]

Write the power series for \exp(i z)

The power series definition for \exp(i z) is: \[ e^{i z} = \sum_{n=0}^{\infty} \frac{(i z)^n}{n!} \]

Simplify the power series for \exp(i z)

Rewrite the power series for \exp(i z) by separating terms involving even and odd powers of z: \[ e^{i z} = \sum_{n=0}^{\infty} \frac{(i z)^{2n}}{(2n)!} + \sum_{n=0}^{\infty} \frac{(i z)^{2n+1}}{(2n+1)!} \]

Substitute i^n terms

Recall that \i^{2n} = (-1)^n and \i^{2n+1} = (-1)^n i. Substitute these into the series: \[ e^{i z} = \sum_{n=0}^{\infty} (-1)^n \frac{z^{2n}}{(2n)!} + i \sum_{n=0}^{\infty} (-1)^n \frac{z^{2n+1}}{(2n+1)!} \]

Compare with power series definitions for \sin z and \cos z

Notice that these series match the power series definitions for \cos z and \sin z. Therefore, we can write: \[ e^{i z} = \cos z + i \sin z \]

Key concepts

Power Series

Power series are a way of representing functions as an infinite sum of terms. Each term is composed of a coefficient and a variable raised to a power. The general form of a power series is:
\[\begin{equation} \text{Power Series} = \text{Sum} \big( a_n \frac{z^n}{n!} \big) \end{equation}\]
where:

• ### a_n - coefficients that can depend on n, ###
• ### z - variable, ###
• ### n - non-negative integer. ###

Power series make it easier to understand and manipulate complex functions. For example, we can define functions like sine, cosine, and the exponential function using their power series representations. These series help us to analyze and calculate these functions, especially in complex analysis.

Exponential Function

The exponential function, often written as exp(x) or e^x, is a fundamental mathematical function with widespread applications. For real numbers, it is defined as:
\[\begin{equation} e^x = \text{Sum} \big( \frac{x^n}{n!} \big) \end{equation}\]
When we extend it to complex numbers, we use the same definition. But interestingly, it brings out beautiful relationships with trigonometric functions. For a complex number z = x + i y, the power series definition of the exponential function is:
\[\begin{equation} e^{i z} = \text{Sum} \big( \frac{(i z)^n}{n!} \big) \end{equation}\]
If you expand this power series, you'll notice patterns that link directly to \(\text{cosine and sine functions}\). Understanding the exponential function in terms of power series is crucial for grasping concepts in complex analysis and other advanced mathematics areas.

Sine and Cosine Functions

The sine and cosine functions are essential in trigonometry and are defined using their power series representations. The power series definitions for sine and cosine are:
\[\begin{equation} \text{sin}(z) = \text{Sum} \big( (-1)^n \frac{z^{2n+1}}{(2n+1)!} \big) \end{equation}\]
\[\begin{equation} \text{cos}(z) = \text{Sum} \big( (-1)^n \frac{z^{2n}}{(2n)!} \big) \end{equation}\]
These series converge for all complex numbers z. By expanding these series, we can see how sine and cosine functions oscillate. It's fascinating that their definitions involve patterns of positive and negative signs and factorials, making them beautiful examples of infinite series.

Euler's Formula

Euler's Formula is one of the most elegant results in mathematics, connecting the exponential function with sine and cosine functions. It states:
\[\begin{equation} e^{i z} = \text{cos}(z) + i \text{sin}(z) \end{equation}\]
We derive this formula by comparing the power series of \(\text{exp(i z)}\), \(\text{sin(z)}\), and \(\text{cos(z)}\). When we expand the power series for \(\text{exp(i z)}\), we get:
\[\begin{equation} e^{i z} = \text{Sum} \big( \frac{(i z)^n}{n!} \big) = \text{Sum} \big( (-1)^n \frac{z^{2n}}{(2n)!} \big) + i \text{Sum} \big( (-1)^n \frac{z^{2n+1}}{(2n+1)!} \big) \end{equation}\]
You will notice that the first sum represents the cosine function and the second sum with i represents the sine function. This relationship is valuable in both theoretical and applied mathematics, especially in fields like electrical engineering, quantum mechanics, and signal processing.