Answer
Verified
410.7k+ views
Hint: In this question we will use the property of complex numbers and use logarithm to simplify the equations and apply the condition as given in the question. Compare the imaginary parts separately to get the accurate answer.
Complete step by step solution:
As we know that a complex number can be expressed as $x + iy$, here $x$ is the real term and $y$ is the imaginary term.
According to the question it is given that,
${i^{\left( {\alpha + i\beta } \right)}} = \alpha + i\beta $ ………...….. (1)
Take log on both sides in the equation (1).
$\left( {\alpha + i\beta } \right)\log i = \log \left( {\alpha + i\beta } \right)$……..……. (2)
Apply the condition of ${\log _e}\left( {x + iy} \right) = {\log _e}\left( {r{e^{i\theta }}} \right)$ in the equation (2).
Here, $r$ is the modulus value or magnitude value of $x + iy$ and $\theta $ be the argument of $x + iy$.
So,
$\log \left( {\alpha + i\beta } \right) = \left( {\alpha + i\beta } \right)\left[ {i\dfrac{\pi }{2} + 2\pi i} \right],n\varepsilon I$……………...(3)
Now, apply the formula of $\log \left( {\alpha + i\beta } \right)$n equation (3) and multiply by $i$ on both sides.
$
\dfrac{1}{2}\log \left( {{\alpha ^2} + {\beta ^2}} \right) + i{\tan ^{ - 1}}\dfrac{\beta }{\alpha } = i\left( {\alpha + i\beta } \right)\left( {4n + 1} \right)\dfrac{\pi }{2} \\
\dfrac{1}{2}\log \left( {{\alpha ^2} + {\beta ^2}} \right) + i{\tan ^{ - 1}}\dfrac{\beta }{\alpha } = i\alpha \left( {4n + 1} \right)\dfrac{\pi }{2} - \beta \left( {4n + 1} \right)\dfrac{\pi }{2}
$.........……..(4)
Compare the imaginary parts in the equation (4) and it is written as,
$
\dfrac{1}{2}\log \left( {{\alpha ^2} + {\beta ^2}} \right) = - \dfrac{\beta }{2}\left( {4n + 1} \right) \\
$
Therefore, ${\alpha ^2} + {\beta ^2} = {e^{ - \left( {4n + 1} \right)\pi \beta }}$
Hence, the correct option is D.
Note: As we know that the complex numbers are the combination of real and imaginary parts which can be expressed as $x + iy$. Complex numbers enlarge the concept of $1 - D$ number line to $2 - D$ complex plane. The concept of complex numbers is widely used in the field of mathematics and physics. If the real part of the complex number is zero then the complex number is purely imaginary and if the imaginary part of the complex number is zero then the complex number is purely real.
Complete step by step solution:
As we know that a complex number can be expressed as $x + iy$, here $x$ is the real term and $y$ is the imaginary term.
According to the question it is given that,
${i^{\left( {\alpha + i\beta } \right)}} = \alpha + i\beta $ ………...….. (1)
Take log on both sides in the equation (1).
$\left( {\alpha + i\beta } \right)\log i = \log \left( {\alpha + i\beta } \right)$……..……. (2)
Apply the condition of ${\log _e}\left( {x + iy} \right) = {\log _e}\left( {r{e^{i\theta }}} \right)$ in the equation (2).
Here, $r$ is the modulus value or magnitude value of $x + iy$ and $\theta $ be the argument of $x + iy$.
So,
$\log \left( {\alpha + i\beta } \right) = \left( {\alpha + i\beta } \right)\left[ {i\dfrac{\pi }{2} + 2\pi i} \right],n\varepsilon I$……………...(3)
Now, apply the formula of $\log \left( {\alpha + i\beta } \right)$n equation (3) and multiply by $i$ on both sides.
$
\dfrac{1}{2}\log \left( {{\alpha ^2} + {\beta ^2}} \right) + i{\tan ^{ - 1}}\dfrac{\beta }{\alpha } = i\left( {\alpha + i\beta } \right)\left( {4n + 1} \right)\dfrac{\pi }{2} \\
\dfrac{1}{2}\log \left( {{\alpha ^2} + {\beta ^2}} \right) + i{\tan ^{ - 1}}\dfrac{\beta }{\alpha } = i\alpha \left( {4n + 1} \right)\dfrac{\pi }{2} - \beta \left( {4n + 1} \right)\dfrac{\pi }{2}
$.........……..(4)
Compare the imaginary parts in the equation (4) and it is written as,
$
\dfrac{1}{2}\log \left( {{\alpha ^2} + {\beta ^2}} \right) = - \dfrac{\beta }{2}\left( {4n + 1} \right) \\
$
Therefore, ${\alpha ^2} + {\beta ^2} = {e^{ - \left( {4n + 1} \right)\pi \beta }}$
Hence, the correct option is D.
Note: As we know that the complex numbers are the combination of real and imaginary parts which can be expressed as $x + iy$. Complex numbers enlarge the concept of $1 - D$ number line to $2 - D$ complex plane. The concept of complex numbers is widely used in the field of mathematics and physics. If the real part of the complex number is zero then the complex number is purely imaginary and if the imaginary part of the complex number is zero then the complex number is purely real.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE
Which are the Top 10 Largest Countries of the World?
How do you graph the function fx 4x class 9 maths CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Why is there a time difference of about 5 hours between class 10 social science CBSE