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Markov2
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Solve ${{\left( \dfrac{1+i}{\sqrt{2}} \right)}^{x}}+{{\left( \dfrac{1-i}{\sqrt{2}} \right)}^{x}}=\sqrt{2}.$
Is there a faster way to solve this?
Is there a faster way to solve this?
Last edited:
Markov said:Solve ${{\left( \dfrac{1+i}{\sqrt{2}} \right)}^{x}}+{{\left( \dfrac{1-i}{\sqrt{2}} \right)}^{x}}=\sqrt{2}.$
Is there a faster way to solve this?
Markov said:Oh yes, you just used Euler's formula, so the solutions are $\dfrac{\pi }{4}ix = \dfrac{\pi }{4} \pm 2k\pi ,{\text{ }}k \in \mathbb{Z}$ and $\dfrac{\pi }{4}ix = \dfrac{{7\pi }}{4} \pm 2k\pi ,{\text{ }}k \in \mathbb{Z},$ are those correct?
Thanks for the help!
I am a bit confused by the replies.Markov said:Solve ${{\left( \dfrac{1+i}{\sqrt{2}} \right)}^{x}}+{{\left( \dfrac{1-i}{\sqrt{2}} \right)}^{x}}=\sqrt{2}.$
Then might look at $x=\pm 7,~\pm 9,~\pm 23,~\pm 25,\cdots$.Markov said:No, $x$ is supposed to be real, thanks for the catch Plato!
A complex equation is a mathematical expression that contains both real and imaginary numbers. It is typically in the form of a + bi, where a is the real part and bi is the imaginary part.
The purpose of solving complex equations is to find the values of the variables that make the equation true. This can help solve problems in various fields such as engineering, physics, and economics.
To solve a complex equation, you need to isolate the variable on one side of the equation and simplify the other side. This can be done by using algebraic operations such as addition, subtraction, multiplication, and division, as well as complex number operations such as conjugation and multiplication by the complex conjugate.
The common methods for solving complex equations include substitution, elimination, and graphing. These methods can be used for both linear and quadratic complex equations.
Complex equations have a wide range of applications in various fields such as electrical engineering, signal processing, quantum mechanics, and control theory. They are also used in solving problems involving alternating currents, oscillatory motion, and resonance.