Question

Find one or more values of each of the following complex expressions and compare with a computer solution. $$(-e)^{i \pi}$$

Short answer

The value of \[(-e)^{i\pi}\] is approximately \(5.242e^{-5}\).

Step-by-step solution

Understand the Expression

The expression \[(-e)^{i\pi}\] involves a negative base raised to a complex exponent. Note that the base here is the exponential function 'e', which is approximately equal to 2.718.

Express the Base in Polar Form

To handle the negative base, convert \(-e\) into its polar form. The magnitude of \(-e\) is \(|-e| = e\). The angle, or argument, of \(-e\) is \(\pi + 0 = \pi\) radians (since \(e\) lies on the negative x-axis). Therefore, \(-e = e^{i\pi}\).

Apply the Exponent

Use the property of exponents \((a^b)^c = a^{bc}\) to simplify the expression: \[(-e)^{i\pi} = (e^{i\pi})^{i\pi}\] simplifies to \[e^{i\pi \cdot i\pi} = e^{-\pi^2}\]

Compute the Value

Since \(e\) raised to any real number can be computed directly, the value of \[e^{-\pi^2} = e^{-9.869}\approx 5.242e^{-5}\]

Compare with a Computer Solution

A computer algebra system (CAS) or scientific calculator can be used to verify that \[(-e)^{i\pi}\] indeed simplifies to approximately \(5.242e^{-5}\).

Key concepts

Polar Form

To work with complex numbers effectively, converting them to polar form is essential. The polar form represents a complex number \, \(z\), using magnitude \, \(r\), and angle \, \(\theta\) \((z=r e^{i \theta})\). This approach is useful as it provides a clear geometric interpretation. For any complex number \, \(z = x + iy\), where \, \(x\) and \, \(y\) are real, the polar form is found as follows:

• Calculate the magnitude \, \(r\) using \, \(|z|=\sqrt{x^2 + y^2}\).
• Determine the argument \, \(\theta\), which is the angle the number makes with the positive x-axis, using \, \(\theta=\arg(z) = \arctan\left(\frac{y}{x}\right)\).

For the given complex number \, \(-e\), the magnitude \, \(r\) is \(e\) (since magnitudes are always positive), and the argument \, \(\theta\) is \(\pi\) radians (since it lies on the negative x-axis). This gives us \, \(-e = e^{i\pi}\). Understanding this method is crucial for simplifying and solving complex number problems.

Exponential Function

The exponential function is powerful when dealing with complex numbers. Euler's formula, \, \(e^{i \theta} = \cos(\theta) + i\sin(\theta)\), bridges exponential functions and trigonometry. When applied to complex exponents, it allows us to simplify expressions. For example, given a complex exponent \, \(i\pi\), Euler's formula lets us write:

• \(e^{i \theta}\) where \(\theta = \pi\) becomes \(e^{i\pi} = \cos(\pi) + i\sin(\pi)\). Given \(\cos(\pi)=-1\) and \(\sin(\pi)=0\), we get \(e^{i\pi} = -1\).

Combining exponent properties, such as \, \((a^b)^c = a^{bc}\), with complex numbers in the exponential form provides simplification paths. For instance, \, \((-e)^{i\pi} = (e^{i\pi})^{i\pi} = e^{-(\pi^2)}\) results from recognizing patterns in exponential expressions. Mastering these topics gives flexibility in solving many complex number calculations.

Euler's Formula

Euler's Formula, \, \(e^{i \theta} = \cos(\theta) + i\sin(\theta)\), is a cornerstone of complex number analysis. It connects exponential functions to trigonometric functions, simplifying many complex calculations. For any angle \, \(\theta\), Euler’s formula expresses the complex number on the unit circle in terms of its cosine and sine components. Key Interpretations:

• For \, \(\theta = \pi\), the formula becomes \(e^{i\pi} = -1\). This sophisticated relationship underpins many trigonometric identities and simplifies complex arithmetic.
• In solving \, \((-e)^{i\pi}\), converting \, \(-e\) into polar form using Euler’s formula is necessary. Recognizing \, \(-e = e^{i\pi}\) indicates utilizing \(e^{i\pi} = -1\).

Understanding Euler's Formula is essential for moving between exponential and trigonometric forms. It not only aids in solving exercises like \, \((-e)^{i\pi}\), but also in grasping broader mathematical concepts involving periodicity, wave functions, and signal processing.