Question

Find the real and imaginary parts$\mathit{u}\left(x,y\right)$ and$\mathbf{v}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)}$ of the following functions.

$\overline{{\mathbf{e}}^{\mathbf{z}}}$

Short answer

The real part$u\left(x,y\right)$ of the function is $e^x\cos y$and the imaginary part$v\left(x,y\right)$of the function is $e^x\sin y$.

Step-by-step solution

Definition of complex number and complex conjugate

• Complex numberis represented by $a+ib$, where a is the real number and b is the imaginary number.
• Complex conjugateof a complex number which has the same real part and the same imaginary part but with opposite sign. Complex conjugate of complex number z is represented by$\overline{z}$ .

For example: complex conjugate of a complex number$3+2i$ is$3-2i$ .

Solve complex number

Given the function is $\overline{e^z}$.

The complex number $$" width="9">

$z=x+iy$, where x is a real part and y is an imaginary part.

Substitute the complex number and simplify.

$$\begin{gathered} \overline{e^z}=\overline{e^{\left(x+iy\right)}} \\ =\overline{e^x·e^{iy}} \end{gathered}$$

Use Euler’s formula, $e^{ix}=\cos x+i\sin x$and simplify.

$$\begin{gathered} \overline{e^z}=\overline{e^x·e^{iy}} \\ =\overline{e^x\left(cosy+isiny\right)} \\ =\overline{\left(e^{x}cosy+i{e}^{x}siny\right)} \\ =e^{x}cosy-i{e}^{x}siny \end{gathered}$$

Find real and imaginary parts

Simplify the expression future.

$$\begin{gathered} \overline{e^z}=\overline{e^x·e^{iy}} \\ =\overline{e^x\left(\cos y+i\sin y\right)} \\ =\overline{\left(e^x\cos y+i{e}^x\sin y\right)} \\ =e^x\cos y-i{e}^x\sin y \end{gathered}$$

Hence the real part is $e^x\cos y$and imaginary part is $e^x\cos y$.