Answer
423.9k+ views
Hint: express \[i\] in Euler form. The Euler formula is the way to express complex quantities into trigonometric ratios, In particular sin and cos ratios. If we are going into trigonometry then $\theta$ will come into play.
Complete step by step solution: Given \[{i^i}\], where \[i = \sqrt { - 1} \]
We know \[i = \cos \dfrac{\pi }{2} + i\sin \dfrac{\pi }{2} = {e^{i\dfrac{\pi }{2}}}\] So the value of \[{i^i}\]will be
\[\begin{gathered}
\therefore {i^i} = {\left( {{e^{i\dfrac{\pi }{2}}}} \right)^i} \\
= {e^{{i^2}\dfrac{\pi }{2}}} \\
= {e^{ - \pi /2}} \approx 0.20787957635076193 \\
\end{gathered} \]
General solution:
\[\begin{gathered}
i = {e^{i(2n\pi + \dfrac{\pi }{2})}} \\
= {e^{i(4n + 1)\dfrac{\pi }{2}}} \\
\end{gathered} \]
\[\begin{gathered}
\therefore {i^i} = {\left( {{e^{i\dfrac{{(4n + 1)\pi }}{2}}}} \right)^i} \\
= {e^{{i^2}\dfrac{{(4n + 1)\pi }}{2}}} \\
= {e^{ - \dfrac{{(4n + 1)\pi }}{2}}} \\
\end{gathered} \]
Where, n is any integer i.e. n=0, ±1, ±2, ±3,… n = 0, ±1, ±2, ±3,…
Now, substituting the integral values of n, one should get infinitely many solutions as follows
\[{i^i} = \ldots {e^{ - \dfrac{{9\pi }}{2}}},{e^{ - \dfrac{{5\pi }}{2}}},{e^{ - \dfrac{\pi }{2}}},{e^{\dfrac{{3\pi }}{2}}},{e^{\dfrac{{7\pi }}{2}}}...\]
Additional solution: Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for any real number x:
where e is the base of the natural logarithm, i is the imaginary unit, and cos and sin are the trigonometric functions cosine and sine respectively. This complex exponential function is sometimes denoted cis x ("cosine plus i sine"). The formula is still valid if x is a complex number, and so some authors refer to the more general complex version as Euler's formula.
Euler's formula is ubiquitous in mathematics, physics, and engineering. The physicist Richard Feynman called the equation "our jewel" and "the most remarkable formula in mathematics".
When , Euler's formula evaluates to , which is known as Euler's identity.
Note: \[{i^i}\] has an infinite number of solutions. All the solutions contain positive value. Exponentiating this equation yields Euler's formula. Note that the logarithmic statement is not universally correct for complex numbers, since a complex logarithm can have infinitely many values, differing by multiples of $2i\pi$.
Complete step by step solution: Given \[{i^i}\], where \[i = \sqrt { - 1} \]
We know \[i = \cos \dfrac{\pi }{2} + i\sin \dfrac{\pi }{2} = {e^{i\dfrac{\pi }{2}}}\] So the value of \[{i^i}\]will be
\[\begin{gathered}
\therefore {i^i} = {\left( {{e^{i\dfrac{\pi }{2}}}} \right)^i} \\
= {e^{{i^2}\dfrac{\pi }{2}}} \\
= {e^{ - \pi /2}} \approx 0.20787957635076193 \\
\end{gathered} \]
General solution:
\[\begin{gathered}
i = {e^{i(2n\pi + \dfrac{\pi }{2})}} \\
= {e^{i(4n + 1)\dfrac{\pi }{2}}} \\
\end{gathered} \]
\[\begin{gathered}
\therefore {i^i} = {\left( {{e^{i\dfrac{{(4n + 1)\pi }}{2}}}} \right)^i} \\
= {e^{{i^2}\dfrac{{(4n + 1)\pi }}{2}}} \\
= {e^{ - \dfrac{{(4n + 1)\pi }}{2}}} \\
\end{gathered} \]
Where, n is any integer i.e. n=0, ±1, ±2, ±3,… n = 0, ±1, ±2, ±3,…
Now, substituting the integral values of n, one should get infinitely many solutions as follows
\[{i^i} = \ldots {e^{ - \dfrac{{9\pi }}{2}}},{e^{ - \dfrac{{5\pi }}{2}}},{e^{ - \dfrac{\pi }{2}}},{e^{\dfrac{{3\pi }}{2}}},{e^{\dfrac{{7\pi }}{2}}}...\]
Additional solution: Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for any real number x:
where e is the base of the natural logarithm, i is the imaginary unit, and cos and sin are the trigonometric functions cosine and sine respectively. This complex exponential function is sometimes denoted cis x ("cosine plus i sine"). The formula is still valid if x is a complex number, and so some authors refer to the more general complex version as Euler's formula.
Euler's formula is ubiquitous in mathematics, physics, and engineering. The physicist Richard Feynman called the equation "our jewel" and "the most remarkable formula in mathematics".
When , Euler's formula evaluates to , which is known as Euler's identity.
Note: \[{i^i}\] has an infinite number of solutions. All the solutions contain positive value. Exponentiating this equation yields Euler's formula. Note that the logarithmic statement is not universally correct for complex numbers, since a complex logarithm can have infinitely many values, differing by multiples of $2i\pi$.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Why Are Noble Gases NonReactive class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
At which age domestication of animals started A Neolithic class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)