Q23P - Mathematical Methods in Physical Sciences

Question

Find the real part, the imaginary part, and the absolute value of 

cosh (x)

Step-by-Step Solution

Answer

The Real part of cosh (ix) is cos(x).

The Imaginary part of cosh (ix) is 0.

The absolute value of cosh (ix) is |cos(x)|.

Step 1: Given Information.

The given expression is cosh (ix).

Step 2: Definition of Complex Numberand Hyperbolic functions.

The domain of a function refers to the range of values that can be plugged into it. This is the set of x values in a function like f (x). The range of a function is the set of values that the function can take. This is the set of values that the function produces when we enter x value. The y values are what you're looking for.

The basic hyperbolic functions are:

1. Hyperbolic sine(sinh)
2. Hyperbolic cosine(cosh)
3. Hyperbolic tangent(tanh)

Step 3: Find the real, imaginary, absolute value of cosh(ix).

$$\begin{gathered} \mathrm{Let} \cosh \left(\mathrm{ix}\right)=\cosh \left(\mathrm{iz}\right). \\ \mathrm{Use} \mathrm{the} \mathrm{formula} \mathrm{mentioned} \mathrm{below}. \\ \sinh \left(\mathrm{iz}\right)=\mathrm{i} \sin \left(\mathrm{z}\right) \\ \tanh \left(\mathrm{iz}\right)=\mathrm{i} \tan \left(\mathrm{z}\right) \\ \mathrm{DivideLeft} \mathrm{Hand} \mathrm{Side} \mathrm{of} \mathrm{first} \mathrm{identity} \mathrm{by} \mathrm{second} \mathrm{identity}. \\ \frac{ \sinh \left(\mathrm{iz}\right)}{\tanh \left(\mathrm{iz}\right)} =\cosh \left(\mathrm{iz}\right) \dots . \left(1\right) \\ \mathrm{Divide} \mathrm{Left} \mathrm{Hand} \mathrm{Side} \mathrm{of} \mathrm{first} \mathrm{identity} \mathrm{by} \mathrm{second} \mathrm{identity}. \dots . \left(2\right) \\ \frac{\mathrm{i} \sinh \left(\mathrm{z}\right)}{\mathrm{i} \tanh \left(\mathrm{z}\right)} =\cosh \left(\mathrm{z}\right) \\ \mathrm{Combine} \mathrm{equation} \left(1\right) \mathrm{and} \mathrm{equation} \left(2\right). \\ \cosh \left(\mathrm{iz}\right)=\cosh \left(\mathrm{z}\right) \\ \cosh \left(\mathrm{ix}\right)=\cosh \left(\mathrm{x}\right) \\ \mathrm{Hence}, \\ \mathrm{The} \mathrm{Real} \mathrm{part} \mathrm{of} \cosh \left(\mathrm{ix}\right) \mathrm{is} \cosh \left(\mathrm{x}\right) . \\ \mathrm{The} \mathrm{Imaginary} \mathrm{part} \mathrm{of} \cosh \left(\mathrm{ix}\right) \mathrm{is} 0. \\ \mathrm{The} \mathrm{absolute} \mathrm{value} \mathrm{of} \cosh \left(\mathrm{ix}\right) \mathrm{is} \cosh \left(\mathrm{x}\right). \end{gathered}$$