Complex Algebra: Trigonometry (tan)

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Homework Help Overview

The discussion revolves around the complex extension of the tangent function, specifically exploring the values in the complex plane for which the magnitude of tan(q) is infinite. Participants are attempting to understand the implications of this condition and how it relates to the poles of the function.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • Participants are examining the expression of tan(q) in terms of complex exponentials and questioning how to identify values of q that lead to |tan(q)| = ∞. There is discussion about the conditions under which the denominator vanishes and the implications of finding specific values like q = π/2.

Discussion Status

Some participants have identified potential solutions and are exploring the meaning of these values in the context of the problem. There is an ongoing inquiry into the range of values for q and how they relate to the identified poles. The discussion reflects a mix of understanding and confusion, with no clear consensus yet reached.

Contextual Notes

Participants are grappling with the complexity of the problem, noting that it appears simpler than other topics they have encountered, yet it presents unique challenges. There is mention of the need to consider integer multiples in the context of complex exponentials, indicating a broader range of solutions may exist.

PedroB
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Homework Statement



Let tan(q) with q ε ℂ be defined as the natural extension of tan(x) for real values

Find all the values in the complex plane for which |tan(q)| = ∞

Homework Equations



Expressing tan(q) as complex exponentials:

(e^iq - e^(-iq))/i(e^iq + e^(-iq))

The Attempt at a Solution



I really have no idea how to get around this problem. No matter what I equate 'q' to I don't seem to get a valid answer. Any help would be greatly appreciated.
 
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PedroB said:

Homework Statement



Let tan(q) with q ε ℂ be defined as the natural extension of tan(x) for real values

Find all the values in the complex plane for which |tan(q)| = ∞

Homework Equations



Expressing tan(q) as complex exponentials:

(e^iq - e^(-iq))/i(e^iq + e^(-iq))

The Attempt at a Solution



I really have no idea how to get around this problem. No matter what I equate 'q' to I don't seem to get a valid answer. Any help would be greatly appreciated.

tan(q) having a pole means the denominator must vanish. So you want to find values of q where e^(iq)+e^(-iq)=0. Multiply both sides by e^(iq).
 
Ok, so I get q= ∏/2 (which makes sense since tan(∏/2)= ∞), but what does this mean exactly? Are the range of values when x + iy = ∏/2? Surely this simply leads to x = ∏/2 which is evidently not a 'range' of values. This is the simpler of problems that I need to do, though it's the one that's given me more trouble (I've got a few complex derivatives questions which were pretty straight forward to me compared to this, strangely enough)
 
PedroB said:
Ok, so I get q= ∏/2 (which makes sense since tan(∏/2)= ∞), but what does this mean exactly? Are the range of values when x + iy = ∏/2? Surely this simply leads to x = ∏/2 which is evidently not a 'range' of values. This is the simpler of problems that I need to do, though it's the one that's given me more trouble (I've got a few complex derivatives questions which were pretty straight forward to me compared to this, strangely enough)

e^(x)=e^(x+2*i*pi*n) for any integer n. So, yes, there are lots of solutions for q.
 

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