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Even and odd functions form an important part of function analysis, helping in decomposing complex functions into simpler parts. An even function remains unchanged if you replace its variable with its negative counterpart, satisfying the property: \[ f(x) = f(-x) \] For example, the function \( f(x) = x^2 \) is even since \( f(x) = (x)^2 = (-x)^2 \).

An odd function, on the other hand, changes sign when you replace its variable with its negative counterpart and satisfies: \[ f(x) = -f(-x) \] For example, the function \( f(x) = x^3 \) is odd since \( f(x) = (x)^3 = -(-x^3) \). Decomposing a function into even and odd parts involves using the formulas: \[ f_{even}(x) = \frac{f(x) + f(-x)}{2} \] and \[ f_{odd}(x) = \frac{f(x) - f(-x)}{2} \].

By applying these formulas, we can rewrite any function as the sum of even and odd components, providing a deeper understanding and simplicity in further calculations.

Complex exponentials involve functions that involve complex numbers, typically in the form of \( e^{i \theta} \), where \( i \) is the imaginary unit and \( \theta \) is a real number. Euler's formula provides a crucial relationship that helps decompose complex exponentials: \[ e^{i \theta} = \text{cos}(\theta) + i \text{sin}(\theta) \]. This formula shows that a complex exponential can be broken down into cosine and sine functions, which are purely real and imaginary parts.

To decompose \( e^{i n x} \) into even and odd functions, we use the following breakdown:

• Even part: \[ f_{even}(x) = \frac{e^{i n x} + e^{-i n x}}{2} = \text{cos}(n x) \]
• Odd part: \[ f_{odd}(x) = \frac{e^{i n x} - e^{-i n x}}{2} = i \text{sin}(n x) \]

Thus, combining these parts, we get: \[ e^{i n x} = \text{cos}(n x) + i \text{sin}(n x) \]. This decomposition simplifies complex exponentials into more manageable real and imaginary parts.

Hyperbolic functions are analogs of the trigonometric functions but for a hyperbola rather than a circle. They are defined using exponential functions:

• Hyperbolic cosine: \[\text{cosh}(x) = \frac{e^x + e^{-x}}{2} \]
• Hyperbolic sine: \[\text{sinh}(x) = \frac{e^x - e^{-x}}{2} \]

These functions appear frequently in various areas of mathematics and physics.

Decomposing the function \( t e^x \) into even and odd parts, we use the definitions of hyperbolic functions:

• Even part: \[ f_{even}(x) = \frac{t e^x + t e^{-x}}{2} = t \text{cosh}(x) \]
• Odd part: \[ f_{odd}(x) = \frac{t e^x - t e^{-x}}{2} = t \text{sinh}(x) \]

Therefore, the decomposition is: \[ t e^x = t \text{cosh}(x) + t \text{sinh}(x) \]. This relationship reflects how exponentials and hyperbolic functions connect and how they can break down into simpler even and odd components.