Q18P - Mathematical Methods in Physical Sciences

Question

Show that tanh z never takes the values $\mathbf{\pm }\mathbf{1}$.

Step-by-Step Solution

Answer

It has been proved that tanh z never takes the values $\pm 1$.

Step 1: Given Information.

The given expression is tanh z.

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: Solve the equation .

Solve the given equation tan z = i in the hint.

$$\begin{gathered} \frac{\left[e^{\left(z\right)}-e^{\left(-z\right)}\right]}{\left[e^{\left(z\right)}+e^{\left(-z\right)}\right]}=-1 \\ e^{\left(z\right)}-e^{\left(-z\right)}=-e^{\left(z\right)}-e^{\left(-z\right)} \\ 2e^{\left(z\right)}=0 \end{gathered}$$

Previous equation has no exact solution because $e^{\left(z\right)}$ is zero only at $-\infty$ .

$$\begin{gathered} \frac{\left[e^{\left(z\right)}-e^{\left(-z\right)}\right]}{\left[e^{\left(z\right)}+e^{\left(-z\right)}\right]}=1 \\ e^{\left(z\right)}-e^{\left(-z\right)}=-e^{\left(z\right)}-e^{\left(-z\right)} \\ -2e^{\left(-z\right)}=0 \end{gathered}$$

Previous equation has no exact solution because $e^{\left(z\right)}$ is zero only at $-\infty$.

 

The solution of the equation is $\pm \infty$ which is not possible.