Complex Variables Algebra Solutions / Argument/Modulus

In summary, the conversation discusses solving for a in the equation 2\log(a^2 - 1) = \pi i, where a is a complex variable. The attempt at a solution involves expanding the logarithm and using properties of complex numbers, but leads to a complicated answer. An alternative approach is suggested, involving equating real and imaginary parts and considering the geometric interpretation.
  • #1
gbu
9
0

Homework Statement



Solve for a, [itex] a \in \mathbb{C}[/itex]

[itex]
\frac{2\ln(a^2 - 1)}{\pi i} = 1
[/itex]

Homework Equations


N/A.

The Attempt at a Solution



Reorganizing the equation.
[itex]2\log(a^2 - 1) = \pi i[/itex]
 
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  • #2
gbu said:

Homework Statement



Solve for a, [itex] a \in \mathbb{C}[/itex]

[itex]
\frac{2\log(a^2 - 1)}{\pi i} = 1
[/itex]

Homework Equations


N/A.

The Attempt at a Solution



Reorganizing the equation.
[itex]2\log(a^2 - 1) = \pi i[/itex]
Expanding the logarithm.
[itex]2\ln(|a^2 - 1|) + i \textrm{arg}(a^2 - 1) = \pi i[/itex]I think what I'm stuck on is that I don't know how to evaluate my length/argument of an arbitrary complex variable like that. I know how to solve them if I'm given a value of a (a = x + iy, then [itex]|a| = \sqrt(x^2 + y^2)[/itex] and [itex]arg(a) = \tan^{-1} \frac{y}{x}[/itex]), but without the value of a I'm not sure where to go.

so a = x+yi. Then (x+yi)^2-1 = (x^2-y^2-1)+2xyi. What is the modulus of that?

Put your i's together as well.
 
  • #3
It's

[itex]
\sqrt{(4x^2 y^2 + (x^2-y^2-1)^2)}
[/itex]

Which I suppose gets me to an answer, but its certainly not a pretty one. Wolfram Alpha gives a very simple answer to the question (a^2 = sqrt(1+i))
 
  • #4
gbu said:
It's

[itex]
\sqrt{(4x^2 y^2 + (x^2-y^2-1)^2)}
[/itex]

Which I suppose gets me to an answer, but its certainly not a pretty one. Wolfram Alpha gives a very simple answer to the question (a^2 = sqrt(1+i))

There may be a better way to do this one. Let me think.
 
  • #5
Separate to log = pi i/2 and raise use the inverse function e.
 
  • #6
Look at what the parts of
[itex] 2\ln(|a^2 - 1|) + 2 i \arg(a^2 - 1) = \pi i [/itex]
are telling you. Equate real and imaginary parts, so [itex]\ln(|a^2-1|)=0 [/itex] and [itex]\arg(a^2 - 1)= \frac{\pi}{2}[/itex]. Without solving any equations and just thinking geometrically, what do those two things tell you about [itex]a^2-1[/itex]?
 

1. What is the definition of a complex variable?

A complex variable is a quantity that has both a real and an imaginary component. It is typically represented in the form of a + bi, where a is the real part and bi is the imaginary part.

2. What is the argument of a complex number?

The argument of a complex number is the angle between the positive real axis and the vector representing the complex number in the complex plane. It is typically denoted by the symbol theta (θ).

3. How do you find the modulus of a complex number?

The modulus of a complex number is the distance from the origin to the point representing the complex number in the complex plane. It can be found using the Pythagorean theorem, where the real and imaginary parts are the legs of a right triangle and the modulus is the hypotenuse.

4. What are the properties of complex variables?

Some of the properties of complex variables include commutativity, associativity, distributivity, and the existence of additive and multiplicative inverses. They also follow the laws of exponents and logarithms, and can be represented using polar form.

5. How are complex variables used in applications?

Complex variables are used in a variety of applications, such as in engineering, physics, and mathematics. They are particularly useful in solving differential equations, analyzing signals and systems, and understanding the behavior of electric circuits and mechanical systems.

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