Question

Show that for any real \(y,\left|e^{i y}\right|=1 .\) Hence show that \(\left|e^{z}\right|=e^{x}\) for every complex \(z\).

Short answer

For any real \( y \), \( |e^{iy}| = 1 \) and for any complex \( z \), \( |e^{z}| = e^{x} \).

Step-by-step solution

- Recall Euler's Formula

According to Euler's formula, for any real number y, we have \[ e^{iy} = \text{cos}(y) + i \text{sin}(y) \].

- Calculate the Magnitude

To find the magnitude of \( e^{iy} \), use the formula for the magnitude of a complex number: \[ |a + ib| = \text{sqrt}(a^2 + b^2) \]. Thus, \( |e^{iy}| = \text{sqrt}(\text{cos}^2(y) + \text{sin}^2(y)) \).

- Use the Pythagorean Identity

Recognize that \( \text{cos}^2(y) + \text{sin}^2(y) = 1 \) by the Pythagorean identity. Therefore, \( |e^{iy}| = \text{sqrt}(1) = 1 \). This proves that \( |e^{iy}| = 1 \).

- Express a Complex Number

Consider any complex number \( z = x + iy \), where \( x \) and \( y \) are real numbers. Then, \( e^{z} = e^{x + iy} = e^{x} \times e^{iy} \).

- Find the Magnitude of the Complex Exponential

Using the property of magnitudes, \( |e^{z}| = |e^{x} \times e^{iy}| = |e^{x}| \times |e^{iy}| \).

- Simplify Using Previous Results

Since \( |e^{iy}| = 1 \) (from Step 3) and \( |e^{x}| = e^{x} \) (as \( e^{x} \) is real and positive), we get \( |e^{z}| = e^{x} \).

Key concepts

Euler's Formula

Euler's formula is a key concept in understanding complex numbers and their exponential form. It states that for any real number \( y \), the complex exponential \( e^{iy} \) can be expressed as \[ e^{iy} = \text{cos}(y) + i \text{sin}(y) \]. This formula combines trigonometric functions with complex exponentials, making it easier to work with complex numbers.
By expressing \( e^{iy} \) in terms of sine and cosine, we can understand its behavior and properties better.
For example, the magnitude of a complex number can be directly deduced using this formula.

Complex Numbers

Complex numbers are numbers that have both a real part and an imaginary part. They are written in the form \( a + ib \), where \( a \) is the real part, and \( ib \) is the imaginary part.
The imaginary unit \( i \) is defined by the property \( i^2 = -1 \). Often, complex numbers are used in various fields such as engineering, physics, and mathematics.
They help in representing phenomena that oscillate or wave-like behavior.
When dealing with complex exponentials, Euler's formula proves incredibly useful because it transforms the exponentiation of imaginary numbers into trigonometric terms.

Pythagorean Identity

The Pythagorean identity is a fundamental trigonometric identity and states that \[ \text{cos}^2(y) + \text{sin}^2(y) = 1 \]. This identity is derived from the Pythagorean theorem, which explains the relation between the sides of a right-angled triangle.
When applied to the unit circle, it ensures that the sum of the squares of the cosine and sine of any angle is always one.
We use this identity frequently to simplify the expressions of complex numbers.
In the context of complex exponentials, using Euler's formula, we express \( e^{iy} \) as \( \text{cos}(y) + i \text{sin}(y) \). To find its magnitude, we take the square root of the sum of the squares of the real and imaginary parts, which simplifies beautifully to 1 due to the Pythagorean identity.

Magnitude of Complex Numbers

The magnitude of a complex number, also known as its modulus, is a measure of its size or length in the complex plane. For a complex number \( a + ib \), its magnitude is calculated using the formula\[ |a + ib| = \text{sqrt}(a^2 + b^2) \]. This is derived from the distance formula in a coordinate system.
In the case of the complex exponential \( e^{iy} \), by Euler's formula, we know it is \( \text{cos}(y) + i \text{sin}(y) \).
Using the magnitude formula, we get \[ |e^{iy}| = \text{sqrt}(\text{cos}^2(y) + \text{sin}^2(y)) = \text{sqrt}(1) = 1 \]. Hence, the magnitude of \( e^{iy} \) is always 1.
When dealing with a general complex number \( z = x + iy \), the magnitude of \( e^{z} = e^{x+iy} \) is given by \[ |e^{z}| = e^{x} \times |e^{iy}| \]. Since \( |e^{iy}| = 1 \), we have \[ |e^{z}| = e^{x} \].