Question

Find each of the following in the \(x+i y\) form and check your answers by computer. $$\tanh \frac{3 \pi i}{4}$$

Short answer

The result is \(\tanh\frac{3\text{π}i}{4} = -2i\).

Step-by-step solution

Understanding Hyperbolic Tangent

The hyperbolic tangent function, \(\tanh(z)\), is defined for a complex number \(z\) as \(\tanh(z) = \frac{\text{sinh}(z)}{\text{cosh}(z)}\). We will use this definition to solve the given problem.

Substitute the Given Value

Substitute \(z\) with \(\frac{3\text{π}i}{4}\) into the hyperbolic tangent formula: \(\tanh\frac{3\text{π}i}{4} = \frac{\text{sinh}\frac{3\text{π}i}{4}}{\text{cosh}\frac{3\text{π}i}{4}}\).

Calculate Hyperbolic Sine

Use the definition \(\text{sinh}(z) = \frac{e^z - e^{-z}}{2}\) to find \(\text{sinh}\frac{3\text{π}i}{4}\): \(\text{sinh}\frac{3\text{π}i}{4} = \frac{e^{\frac{3\text{π}i}{4}} - e^{-\frac{3\text{π}i}{4}}}{2}\).

Simplify Hyperbolic Sine

Calculate \(e^{\frac{3\text{π}i}{4}}\) and \(e^{-\frac{3\text{π}i}{4}}\) using Euler's formula \((e^{i\theta} = \text{cos}(\theta) + i \text{sin}(\theta))\): \(e^{\frac{3\text{π}i}{4}} = -\frac{\text{sqrt}(2)}{2} + i \frac{\text{sqrt}(2)}{2}\) and \(e^{-\frac{3\text{π}i}{4}} = -\frac{\text{sqrt}(2)}{2} - i \frac{\text{sqrt}(2)}{2}\). Then substitute into the sinh formula: \(\text{sinh}\frac{3\text{π}i}{4} = \frac{-\frac{\text{sqrt}(2)}{2} + i \frac{\text{sqrt}(2)}{2} + \frac{\text{sqrt}(2)}{2} + i \frac{\text{sqrt}(2)}{2}}{2} = i \text{sqrt(2)}\).

Calculate Hyperbolic Cosine

Use the definition \(\text{cosh}(z) = \frac{e^z + e^{-z}}{2}\) to find \(\text{cosh}\frac{3\text{π}i}{4}\): \(\text{cosh}\frac{3\text{π}i}{4} = \frac{e^{\frac{3\text{π}i}{4}} + e^{-\frac{3\text{π}i}{4}}}{2}\).

Simplify Hyperbolic Cosine

Substitute the previously calculated values: \(\text{cosh}\frac{3\text{π}i}{4} = \frac{-\frac{\text{sqrt}(2)}{2} + i \frac{\text{sqrt}(2)}{2} - \frac{\text{sqrt}(2)}{2} - i \frac{\text{sqrt}(2)}{2}}{2} = -\frac{\text{sqrt}(2)}{2}\).

Compute \(\tanh\frac{3\text{π}i}{4}\)

Divide the results of Step 4 and Step 6: \(\tanh\frac{3\text{π}i}{4} = \frac{i \text{sqrt}(2)}{-\frac{\text{sqrt}(2)}{2}} = -2i\).

Key concepts

hyperbolic tangent

The hyperbolic tangent function, denoted as \(\tanh(z)\), is a fundamental mathematical function that extends the properties of the regular tangent function to hyperbolic geometry. For a complex number \(\tanh(z)\), it is defined as the ratio of the hyperbolic sine (\(\text{sinh}(z)\)) and hyperbolic cosine (\(\text{cosh}(z)\)) functions. The formula is: \[ \tanh(z) = \frac{\text{sinh}(z)}{\text{cosh}(z)} \].
Understanding this relationship is crucial for problems involving complex numbers, as it allows us to transition between different forms and representations smoothly.
In the given exercise, we are asked to find \(\tanh \frac{3\text{π}i}{4}\). By using the definitions of \(\text{sinh}(z)\) and \(\text{cosh}(z)\), we can substitute, simplify, and compute the value step by step.

complex numbers

Complex numbers extend the idea of one-dimensional real numbers to the two-dimensional complex plane by including imaginary components. A complex number is usually written as \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit (\(i^2 = -1\)).
In the exercise, we dealt with complex exponents like \(e^{\frac{3πi}{4}}\). Using Euler's formula, these complex exponentials can be expressed in terms of sines and cosines. For example, \(e^{iθ} = \text{cos}(θ) + i \text{sin}(θ)\) simplifies the problem by converting exponential forms into more manageable trigonometric forms. The same applies even when \(θ\) is a complex number. This is paramount for calculations involving hyperbolic functions defined with complex arguments.

Euler's formula

Euler's formula, named after the Swiss mathematician Leonhard Euler, is one of the most widely celebrated equations in mathematics. It shows a profound relationship between complex exponentials and trigonometric functions: \[ e^{iθ} = \text{cos}(θ) + i \text{sin}(θ) \].
In this exercise, Euler's formula helped simplify the expressions \(e^{\frac{3πi}{4}}\) and \(e^{-\frac{3πi}{4}}\). By converting these exponentials into their sine and cosine components, the problem becomes much more straightforward.
For example: \[ e^{\frac{3πi}{4}} = -\frac{\text{sqrt}(2)}{2} + i \frac{\text{sqrt}(2)}{2} \] and \[ e^{-\frac{3πi}{4}} = -\frac{\text{sqrt}(2)}{2} - i \frac{\text{sqrt}(2)}{2} \]. By using these representations, calculating the hyperbolic sine and cosine functions becomes a matter of simple arithmetic operations on their real and imaginary parts, leading us to the final answer.