Question

$$ \text { Prove that } e^{t_{1}+t_{2}}=e^{t_{1}} e^{t_{2}}\left(z_{1} \in C, z_{2} \in C\right) \text {. } $$

Short answer

Question: Prove that for two complex numbers \(z_1\) and \(z_2\), the equality $$e^{t_1 + t_2} = e^{t_1} e^{t_2}$$ holds, where \(t_1\) and \(t_2\) are complex numbers as well.

Step-by-step solution

Recall the definition of the exponential function for complex numbers

The exponential function for complex numbers is defined in terms of its real and imaginary parts as follows: $$e^z = e^{x + iy} = e^x \cdot e^{iy}$$ where \(z = x + iy\) is a complex number with \(x\) being the real part, \(y\) the imaginary part, and \(i\) the imaginary unit.

Calculate the exponential functions for the sum and product separately

We will now calculate the exponential function for the sum \(t_1 + t_2\) and the product \(e^{t_1} \cdot e^{t_2}\). For the sum: \(e^{t_1 + t_2}\). For the product: \(e^{t_1} \cdot e^{t_2} = \left(e^{x_1 + iy_1}\right)\left(e^{x_2 + iy_2}\right) = e^{x_1} \cdot e^{iy_1} \cdot e^{x_2} \cdot e^{iy_2}\)

Simplify the product expression

By basic algebraic manipulations, $$e^{x_1} \cdot e^{iy_1} \cdot e^{x_2} \cdot e^{iy_2} = e^{x_1 + x_2} \cdot e^{i(y_1 + y_2)}$$

Compare the expressions

The two expressions are: - For the sum: \(e^{t_1 + t_2} = e^{(x_1 + x_2) + i(y_1 + y_2)}\) - For the product: \(e^{t_1} \cdot e^{t_2} = e^{x_1 + x_2} \cdot e^{i(y_1 + y_2)}\) From this comparison, it is observed that $$e^{t_1 + t_2} = e^{x_1 + x_2} \cdot e^{i(y_1 + y_2)} = e^{t_1} \cdot e^{t_2}$$ Hence, the expression is proven to be true for \(z_1\) and \(z_2\) being complex numbers.

Key concepts

Complex Numbers

At the heart of higher mathematics, complex numbers extend the idea of quantity beyond the real numbers used in everyday counting. A complex number is written in the form of \(z = x + iy\), where \(x\) is the real component, \(y\) the imaginary component, and \(i\) is the square root of \( -1 \), often called the imaginary unit. These numbers are pivotal in various fields such as electrical engineering, physics, and among others, in explaining the behavior of exponential functions, as they allow rotations and scaling combined in a single arithmetic operation.

The addition of complex numbers is done by adding their corresponding real and imaginary parts. For multiplication, the distributive property is applied and the fact that \(i^2 = -1\) is used to simplify the products.

Exponential Function Properties

The exponential function, denoted by \(e^x\), where \(e\) is Euler's number, approximately equal to 2.71828, has many unique and useful properties. In the realm of complex numbers, these properties are preserved and include its ability to handle growth and decay models, continuity, and the function's differentiation and integration being proportional to the function itself. When we apply an exponential function to a complex number \(e^{x+iy}\), it can be broken down into its real and imaginary parts as \(e^x\) and \(e^{iy}\), respectively.

The latter part, \(e^{iy}\), can be further represented according to Euler's formula as \(\cos(y) + i\sin(y)\), linking exponential growth to rotational motion on the complex plane. This duality is one of the reasons why exponential functions are so fundamental in scientific calculations that involve periodic or oscillatory behavior.

Algebra of Exponential Functions

When working with exponential functions in algebra, we use several core properties to simplify and manipulate expressions. Key among these properties is the fact that the product of exponential functions with the same base can be simplified by adding their exponents: \(e^{a} \times e^{b} = e^{a+b}\). This law holds true even when dealing with complex exponents, which is central to our current discussion regarding complex numbers.

The provided exercise utilizes this property to show that exponential functions with complex exponents follow the same algebraic rules as those with real exponents. This feature of the exponential function is critical in many areas of mathematics and physics, including the analysis of waveforms and signal processing, where complex exponentials are used to describe waves and oscillations.