Residue of e^(az)/(1+e^z)^2 at I Pi

In summary, the conversation discusses finding the residue of a complex function using Cauchy's kth Integral formula. The person attempted a u substitution but ended up with a residue of 0. They are now considering expanding the function as a Laurent series to find the coefficient of 1/(z-iπ).
  • #1
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Homework Statement



I need to find the residue of e^(az)/(1+e^z)^2 at I Pi. For some reason this is such much harder than I thought it was going to be. Mathematica is not even helping :(.

Homework Equations



Cauchy's kth Integral formula.

The Attempt at a Solution



I made an attempt at doing a u substitution of u=e^z, but I ended up with a residue of 0, which was not what I was expecting. All this lead me to believe that you probably can't use u-substitution strategies with line integrals and expect them to work.
 
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  • #2
Try expanding the function as a Laurent series. Use the fact that

[tex]e^{az} = e^{a(z-i\pi+i\pi)} = e^{ia\pi}e^{a(z-i\pi)}[/tex]

Don't forget to expand both exponentials (in the original function) as series. Remember that all you're interested in is the coefficient of 1/(z-iπ), so just concentrate on the terms that will contribute to that.
 
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Related to Residue of e^(az)/(1+e^z)^2 at I Pi

1. What is the residue of e^(az)/(1+e^z)^2 at I Pi?

The residue of a function at a point is the coefficient of the term with the highest negative power in the Laurent series expansion of the function around that point. In this case, the residue at I Pi would be the coefficient of (z-I Pi)^-1 in the Laurent series expansion of e^(az)/(1+e^z)^2.

2. How do you calculate the residue of e^(az)/(1+e^z)^2 at I Pi?

To calculate the residue at I Pi, you can use the formula Res(f,z) = lim(z->I Pi) [(z-I Pi)f(z)]. In this case, you would need to take the limit as z approaches I Pi of [(z-I Pi)e^(az)/(1+e^z)^2].

3. What is the significance of the residue of e^(az)/(1+e^z)^2 at I Pi?

The residue at a point in a complex function is important because it determines the behavior of the function at that point. In this case, the residue at I Pi would give information about the poles and singularities of the function e^(az)/(1+e^z)^2.

4. Can the residue of e^(az)/(1+e^z)^2 at I Pi be negative?

Yes, the residue of a function at a point can be negative. This would indicate that the function has a pole or singularity at that point. In this case, a negative residue at I Pi would mean that the function e^(az)/(1+e^z)^2 has a pole at I Pi.

5. How can the residue of e^(az)/(1+e^z)^2 at I Pi be used in practical applications?

The residue at a point can be used in many applications, such as in the theory of complex variables and contour integration. It can also be used in solving differential equations and evaluating improper integrals. In this specific case, the residue at I Pi can be used to find the coefficients in the Laurent series expansion of e^(az)/(1+e^z)^2, which can be useful in solving certain types of differential equations.

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