Seemingly easy complex line integral

[Jamin2112]

Homework Statement

Find ∫(e^z+cos(z))/z dz integrated over C₁(0)

Homework Equations

Theorem 6.10 (Cauchy's integral formula)

Let f be analytic in the simply connected domain D and let C be a simple closed positively oriented contour that lies in D. If z₀ is a point that lies interior to C, then

f(z₀) = 1/2πi ∫f(z)/(z-z₀) dz

The Attempt at a Solution

So the answer is 4πi, which is of course what you obtain if you invoke Cauchy's integral formula. But our function isn't analytic inside the region over which we're integrating. (?)http://www.myfacewhen.net/uploads/309-wtf-man.jpg

 

[Dick]

The theorem doesn't say that f(z)/z is analytic inside the contour. If it were the answer would be 0. It's says that f(z) is analytic inside the contour.

 

[Jamin2112]

The theorem doesn't say that f(z)/z is analytic inside the contour. If it were the answer would be 0. It's says that f(z) is analytic inside the contour.

http://pinkie.ponychan.net/chan/files/src/132330639382.gif

























































Dang it. My reading comprehension skills suck.

 

[Jamin2112]

I'll ask another related question just to make this thread worthwhile.

 

[Jamin2112]

Section 7.3, Exercise 9.

Find two Laurent series for z^-1(4-z)^-2 involving powers of z and state where they are valid.

Solution so far. Obviously f(z)=1/(z(4-z)²) has singularities at z = 0 and z = 4. It is therefore analytic in the two annuli A(0,0,4) and A(0,4,∞). In the first case, choose p = 3, since 0 < 3 < 4. By Laurent's theorem, our c_-n will be equal to

1/2πi ∫f(z)/(z-0)^-n+1 dz = 1/2πi ∫dz/(4-z)²z^-n+2
integrated on the curve C₃(0).

But this seems too difficult. There must be a better way. ¿Qué me recomienda Ud.?

 

[Dick]

Section 7.3, Exercise 9.

Find two Laurent series for z^-1(4-z)^-2 involving powers of z and state where they are valid.

Solution so far.


Obviously f(z)=1/(z(4-z)²) has singularities at z = 0 and z = 4. It is therefore analytic in the two annuli A(0,0,4) and A(0,4,∞). In the first case, choose p = 3, since 0 < 3 < 4. By Laurent's theorem, our c_-n will be equal to

1/2πi ∫f(z)/(z-0)^-n+1 dz = 1/2πi ∫dz/(4-z)²z^-n+2
integrated on the curve C₃(0).

But this seems too difficult. There must be a better way. ¿Qué me recomienda Ud.?

You generally find Laurent series by doing things like partial fractions expansions and power series expansions, not by evaluating contour integrals. That would be too difficult. Don't you have any examples showing you these sort of techniques?

 

[Jamin2112]

You generally find Laurent series by doing things like partial fractions expansions and power series expansions, not by evaluating contour integrals. That would be too difficult. Don't you have any examples showing you these sort of techniques?

Yeah, brah. One example is finding the Lauret series for f(z) = 3/(2+z-z²).

f(z) = 3/((1+z)(2-z)) = (by partial fraction decomp.) 1/(1+z) + (1/2)*(1/(1-z/2)). I understand how they found out the power series for (1/2)*(1/(1-z/2)) but not for 1/(1+z). (They just refer me to "Corollary 4.2", which, as far as I can tell, doesn't reveal anything about 1/(1+z))

 

[Dick]

Yeah, brah. One example is finding the Lauret series for f(z) = 3/(2+z-z²).

f(z) = 3/((1+z)(2-z)) = (by partial fraction decomp.) 1/(1+z) + (1/2)*(1/(1-z/2)). I understand how they found out the power series for (1/2)*(1/(1-z/2)) but not for 1/(1+z). (They just refer me to "Corollary 4.2", which, as far as I can tell, doesn't reveal anything about 1/(1+z))

If you know how to power expand 1/(1-z/2), I'm not sure why you would have problem expanding 1/(1+z). Write it as 1/(1-(-z)). Now it looks almost the same as the first.

 

[Jamin2112]

Another quick question:

Correct me if I wrong.

f(z) is analytic in a region D
---> f(z) is continuous in D (converse doesn't necessarily hold),
can be represented as a power series convergent in D,
is differentiable on D, and
is integrable on D.

 

[Jamin2112]

If you know how to power expand 1/(1-z/2), I'm not sure why you would have problem expanding 1/(1+z). Write it as 1/(1-(-z)). Now it looks almost the same as the first.

 

[Dick]

Another quick question:

Correct me if I wrong.

f(z) is analytic in a region D
---> f(z) is continuous in D (converse doesn't necessarily hold),
can be represented as a power series convergent in D,
is differentiable on D, and
is integrable on D.

You aren't just trying to use up your funny graphic things are you? Of course, analytic in D implies differentiable in D. That's what analytic means. And no, it can't necessarily be represented as a single power series valid everywhere in D. Laurent series give you plenty of examples of that. And depends on what you mean by 'integrable'.

 

[Jamin2112]

You aren't just trying to use up your funny graphic things are you? Of course, analytic in D implies differentiable in D. That's what analytic means. And no, it can't necessarily be represented as a single power series valid everywhere in D. Laurent series give you plenty of examples of that. And depends on what you mean by 'integrable'.

(I might use an appropriate graphic if I get the chance)Says Wikipedia:

"In complex analysis, a branch of mathematics, a complex-valued function ƒ of a complex variable z [...] is said to be analytic at a if in some open disk centered at a it can be expanded as a convergent power series"

http://en.wikipedia.org/wiki/Holomorphic_functions_are_analytic

By "integrable" I mean you could integrate it over a curve lying in D. I remember the professor saying something like, "if it's differentiable, it's integrable; and vice versa."

 

[Dick]

(I might use an appropriate graphic if I get the chance)


Says Wikipedia:

"In complex analysis, a branch of mathematics, a complex-valued function ƒ of a complex variable z [...] is said to be analytic at a if in some open disk centered at a it can be expanded as a convergent power series"

http://en.wikipedia.org/wiki/Holomorphic_functions_are_analytic

By "integrable" I mean you could integrate it over a curve lying in D. I remember the professor saying something like, "if it's differentiable, it's integrable; and vice versa."

If it has a convergent power series around a point, then sure it's differentiable and continuous at that point. And also sure, the integral along a curve is well defined. It might be divergent if the curve is unbounded in some way, like having an open endpoint or going to infinity.

 

[Jamin2112]

If it has a convergent power series around a point, then sure it's differentiable and continuous at that point. And also sure, the integral along a curve is well defined. It might be divergent if the curve is unbounded in some way, like having an open endpoint or going to infinity.

Understood, brah. Thanks a million. That's all the questions I have for now; I might hit this place up later tonight, after I work on some practice problems.