- #1

Reshma

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I am trying to teach myself some complex analysis. I am using Complex Numbers by Churchill & Brown as my reference. I have reached the integration section and I am encountering certain difficulties.

For e.g. I have this problem:

[tex]\oint \frac{dz}{z^2 - z -2}, |z|\leq 3[/tex]

I can split up the integrand using partial fractions but I don't know how to interpret the boundary conditions that they have given.

Cauchy's integral formula gives:

[tex]\int_C

\frac{f(z)dz}{z - a} = 2\pi i f(a)[/tex]

Where f(z) is analytic and single valued within a closed curve 'C' and 'a' is any point interior to C.

Cauchy-Goursat theorem states:

[tex]\int_C f(z)dz = \int_a^b f[z(t)] z'(t) dt , a\leq t \leq b [/tex]

I don't know whether I should use the Cauchy-integral formula or the Cauchy-Goursat theorem.

Can someone help me out here?

For e.g. I have this problem:

[tex]\oint \frac{dz}{z^2 - z -2}, |z|\leq 3[/tex]

I can split up the integrand using partial fractions but I don't know how to interpret the boundary conditions that they have given.

Cauchy's integral formula gives:

[tex]\int_C

\frac{f(z)dz}{z - a} = 2\pi i f(a)[/tex]

Where f(z) is analytic and single valued within a closed curve 'C' and 'a' is any point interior to C.

Cauchy-Goursat theorem states:

[tex]\int_C f(z)dz = \int_a^b f[z(t)] z'(t) dt , a\leq t \leq b [/tex]

I don't know whether I should use the Cauchy-integral formula or the Cauchy-Goursat theorem.

Can someone help me out here?

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