Question

Show that (a) \(\tanh \left(\frac{z}{2}\right)=\frac{\sinh x+i \sin y}{\cosh x+\cos y}\), (b) \(\operatorname{coth}\left(\frac{z}{2}\right)=\frac{\sinh x-i \sin y}{\cosh x-\cos y}\).

Short answer

After setting up and simplifying each equation, you'll find that both equations are equivalent to their original definitions, which verifies both identities.

Step-by-step solution

Setup of the tanh proof

Start by expressing \( \tanh(z) \) in exponential form. This is given by \(\frac{e^{z}-e^{-z}}{e^{z}+e^{-z}}\). Now you want to prove \( \tanh \left(\frac{z}{2}\right)=\frac{\sinh x+i \sin y}{\cosh x+\cos y} \). Set \( z = x + iy \) and substitute \( z/2 \) into the expression for \( \tanh(z) \).

Solve for tanh

After substituting \( z/2 \), find the real and imaginary parts of the expression and equate them to the right hand side of the equation. Simplify this so that it matches the original equation.

Setup of the coth proof

Now, prove \( \operatorname{coth}\left(\frac{z}{2}\right)=\frac{\sinh x-i \sin y}{\cosh x-\cos y} \). Begin by writing \( \coth(z) \) in its exponential form, \( \frac{e^{z} + e^{-z}}{e^{z} - e^{-z}} \), and substitute \( z/2 \) in the expression.

Solve for coth

Similar to step 2, find the real and imaginary parts of the expression and equate them to the right hand side of the equation. Simplify it and make sure it matches the original equation.

Key concepts

Tanh and Coth Identities

The hyperbolic functions, tanh and coth, have identities that play an essential part in complex analysis. These identities help express these functions in terms of the variables of real and imaginary parts of complex numbers. This is especially useful when solving equations that involve complex numbers in their trigonometric form.

For the hyperbolic tangent function, or \(\tanh(z)\), there is an identity that allows us to convert it into a form involving sinh and cosh functions. The identity can be expressed as:\

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• \(\tanh(z) = \frac{\sinh(z)}{\cosh(z)}\)

Substituting \(z = x + iy\) and dividing by 2, the function becomes:\
\

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• \(\tanh\left(\frac{z}{2}\right) = \frac{\sinh x + i \sin y}{\cosh x + \cos y}\)

Similarly, the identity for the hyperbolic cotangent function, or \(\coth(z)\), is:

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• \(\coth(z) = \frac{\cosh(z)}{\sinh(z)}\)

By substituting \(z/2\), this becomes:

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• \(\coth\left(\frac{z}{2}\right) = \frac{\sinh x - i \sin y}{\cosh x - \cos y}\)

These identities showcase the relationship between hyperbolic functions and trigonometric functions when dealing with complex numbers. They are fundamental in simplifying and solving complex equations that contain these functions.

Exponential Form of Hyperbolic Functions

Hyperbolic functions can be expressed in exponential form, which reveals their relationship with the exponential function, similar to how trigonometric functions relate to circles. This representation is crucial for understanding and deriving various identities for hyperbolic functions.

The hyperbolic tangent function, \(\tanh(z)\), in exponential form is:

• \(\tanh(z) = \frac{e^{z} - e^{-z}}{e^{z} + e^{-z}}\)

Meanwhile, the hyperbolic cotangent function, \(\coth(z)\), can be expressed as:

• \(\coth(z) = \frac{e^{z} + e^{-z}}{e^{z} - e^{-z}}\)

When the variable \(z\) is replaced with \(z/2\), these expressions help derive the equations shown in the earlier identities.

The exponential forms allow for simplified calculations and algebraic manipulations, making it easier to handle expressions in both theoretical and applied mathematics. Understanding these exponential forms aids greatly in transitioning between complex numbers and their hyperbolic counterparts.

Complex Numbers in Trigonometric Form

Complex numbers are often represented using their trigonometric form, which makes use of a modulus and an argument. This representation is valuable for visualizing and solving problems involving rotations or oscillations in the complex plane.

A complex number \(z = x + iy\) can be expressed in polar form as:

• \(z = r(\cos \theta + i \sin \theta)\)

where \(r\) is the modulus of the complex number, calculated as \(r = \sqrt{x^2 + y^2}\), and \(\theta\) is the argument, often found using \(\tan^{-1}(\frac{y}{x})\).

This trigonometric representation is particularly useful when multiplying or dividing complex numbers. By focusing on the exponential form, \(e^{i\theta}\), it becomes evident that trigonometric identities can be applied to complex expressions. This form connects the study of complex numbers with the geometric interpretations of rotation and scaling, aspects that are critical in numerous fields such as engineering and physics.

Understanding complex numbers through their trigonometric form is integral for evaluating their behavior and interactions especially when used in conjunction with hyperbolic functions.