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Introduction

The csch function is one of the six hyperbolic functions, along with the sinh, cosh, sech, tanh, and coth functions. It is defined as the reciprocal of the hyperbolic sine function, or the inverse hyperbolic sine function. In mathematical notation, the csch function is written as:

csch(x) = 1/sinh(x) = 2/(exp(x) – exp(-x))

where sinh(x) is the hyperbolic sine function, exp(x) is the exponential function, and x is any real number.

Like the other hyperbolic functions, the csch function is important in many areas of mathematics, physics, and engineering. It has a variety of applications in differential equations, calculus, probability theory, and statistics, among other fields.

One of the key properties of the csch function is that it is an odd function, meaning that it satisfies the equation:

csch(-x) = -csch(x)

This means that the csch function is symmetric about the origin, and its graph looks like a downward-opening curve that approaches zero as x approaches positive or negative infinity.

Another important property of the csch function is that it is never equal to zero, except at the point x = 0. This is because the hyperbolic sine function never takes on the value zero, and the reciprocal of a non-zero value is always non-zero.

The csch function is closely related to the sinh function, which is defined as:

sinh(x) = (exp(x) – exp(-x))/2

By taking the reciprocal of both sides of this equation, we get:

csch(x) = 2/(exp(x) – exp(-x)) = 2/sinh(x)

This shows that the csch function is the reciprocal of the sinh function, and can be thought of as a measure of how fast the hyperbolic sine function approaches zero as x approaches infinity.

Like the other hyperbolic functions, the csch function has several useful properties that make it a powerful tool in mathematical analysis. Some of these properties include:

Derivatives: The derivative of the csch function is given by:

d/dx csch(x) = -coth(x) csch(x)

where coth(x) is the hyperbolic cotangent function. This formula can be derived using the quotient rule of calculus, and shows that the csch function is always negative, except at the point x = 0.

Integrals: The integral of the csch function can be found using a technique called partial fraction decomposition. Specifically, we can write:

csch(x) = 2/(exp(x) – exp(-x)) = 2exp(-x)/(1 – exp(-2x))

By making the substitution u = exp(-x), we can rewrite this expression as:

csch(x) = -2/(u^2 – 1) du

which is a standard integral that can be evaluated using trigonometric substitution. The result is:

? csch(x) dx = -ln|coth(x/2)| + C

where C is the constant of integration.

Series expansions: The csch function has a power series expansion, given by:

csch(x) = 1/x – x/3! – 7x^3/5! – 31x^5/7! – …

This series converges for all values of x, and can be used to approximate the value of the csch function to arbitrary precision.

Definition

The hyperbolic cosecant function is defined as the reciprocal of the hyperbolic sine function (sinh):

csch(x) = 1/sinh(x)

The hyperbolic sine function is defined as:

sinh(x) = (e^x – e^(-x))/2

where e is the mathematical constant equal to approximately 2.71828.

Properties

Like other hyperbolic functions, csch has some unique properties that set it apart from trigonometric functions. Here are some of the properties of csch:

1. Domain: The domain of the csch function is the set of all real numbers except for zero.
2. Range: The range of the csch function is the set of all real numbers except for zero.
3. Even and odd functions: The csch function is an odd function, which means that csch(-x) = -csch(x).
4. Asymptotes: The csch function has two asymptotes: x = 0 and y = 0.
5. Periodicity: The csch function is not periodic.

Applications

The hyperbolic cosecant function is used in a variety of fields, including physics, engineering, and mathematics. Here are some of the applications of csch:

1. Electrical engineering: In electrical engineering, the csch function is used to describe the behavior of electrical circuits that contain resistors, capacitors, and inductors. The csch function is particularly useful in analyzing the behavior of AC circuits.
2. Physics: In physics, the csch function is used to describe the behavior of various physical systems, such as vibrating strings and springs. The csch function is also used in quantum mechanics to describe the behavior of particles in a potential well.
3. Mathematics: In mathematics, the csch function is used to solve differential equations that involve hyperbolic functions. The csch function is also used in the study of hyperbolic geometry.

Examples

Example 1: Find the value of csch(3)

Solution:

csch(x) = 1/sinh(x)

csch(3) = 1/sinh(3)

Using the formula for sinh(x), we get:

sinh(3) = (e^3 – e^(-3))/2

sinh(3) = (20.0855 – 0.0498)/2

sinh(3) = 10.0178

Therefore, csch(3) = 1/10.0178 = 0.09982

Example 2: Find the value of x if csch(x) = 5

Solution:

csch(x) = 1/sinh(x)

csch(x) = 5

1/sinh(x) = 5

Solving for sinh(x), we get:

sinh(x) = 1/5

Using the formula for sinh(x), we get:

(e^x – e^(-x))/2 = 1/5

Multiplying both sides by 2, we get:

e^x – e^(-x) = 2/5

Multiplying both sides by e^x, we get:

e^(2x) – 1 = (2/5)e^x

Letting y = e^x, we get:

y^2 – (2/5)y – 1 = 0

Quiz

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