Q13P - Mathematical Methods in Physical Sciences

Question

Find each of the following in the x + iy form and compare a computer solution.

$\mathit{c}\mathit{o}\mathit{s}{\mathit{h}}^{\mathbf{-}\mathbf{1}}\mathbf{(}\mathbf{-}\mathbf{1}\mathbf{)}$

Step-by-Step Solution

Answer

The x + iy form of the given equation ${\cosh }^{-1}(-1) \mathrm{is} {\mathrm{z}}_1=\mathrm{i}(\mathrm{\pi }\pm 2\mathrm{n\pi })$.

Step 1: Given Information.

The given expression is ${\cosh }^{-1}(-1)$.

Step 2: Meaning of rectangular form.

Represent the complex number in rectangular form means writing the given complex number in the form of x + iy in which x is the real part and y is the imaginary part.

Step 3: Convert in quadratic equation.

Consider the complex number $\mathrm{z}={\cosh }^{-1}(-1)$.

Rewrite the above expression.

$\cosh \left(\mathrm{z}\right)=-1$

 

Write the formula for $\sin \left(\theta \right)$.

$\frac{e^{\left(z\right)}+e^{\left(-z\right)}}{2}=-1$

 

Put $e^{\left(zi\right)}=u$

$$\begin{gathered} u+\frac{1}{u}=-2 \\ {u}^2+2u+1=0 \end{gathered}$$ 

Step 4: Solve the quadratic equation.

Write the coefficient and then substitute in the formula.

$$\begin{gathered} a=1 \\ b=2 \\ c=1 \end{gathered}$$

 

Put in the formula.

$$\begin{gathered} u=\frac{-b\pm \sqrt{b^2-4ac}}{2a} \\ {u}_1=\frac{-2\sqrt{4-4}}{2} \\ {u}_1=-1 \end{gathered}$$

Step 5: Convert in rectangular form.

Convert in rectangular form.

 

Find the value of $z_1$.

${\mathrm{z}}_1=\ln \left({\mathrm{u}}_1\right)$

 

Take $n=0.1,2.3,..$. for the values below.

$$\begin{gathered} {\mathrm{z}}_1=\ln (-1) \\ {\mathrm{z}}_1=\ln \left(1\right)+\mathrm{i}\left(\mathrm{\pi }\pm 2\mathrm{n\pi }\right) \\ {\mathrm{z}}_1=\mathrm{i}\left(\mathrm{\pi }\pm 2\mathrm{n\pi }\right) \end{gathered}$$

 

Hence the general solution of the given equation is ${\cosh }^{-1}(-1) \mathrm{is} {\mathrm{z}}_1=\mathrm{i}\left(\mathrm{\pi }\pm 2\mathrm{n\pi }\right)$.