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  • 8.9: Poles
    https://math.libretexts.org/Bookshelves/Analysis/Complex_Variables_with_Applications_(Orloff)/08%3A_Taylor_and_Laurent_Series/8.09%3A_Poles
    Poles refer to isolated singularities.
  • 9.4: Residues
    https://math.libretexts.org/Bookshelves/Analysis/Complex_Variables_with_Applications_(Orloff)/09%3A_Residue_Theorem/9.04%3A_Residues
    In this section we’ll explore calculating residues. We’ve seen enough already to know that this will be useful. We will see that even more clearly when we look at the residue theorem in the next secti...In this section we’ll explore calculating residues. We’ve seen enough already to know that this will be useful. We will see that even more clearly when we look at the residue theorem in the next section.
  • 7.1: Cauchy's Theorem
    https://math.libretexts.org/Bookshelves/Linear_Algebra/Matrix_Analysis_(Cox)/07%3A_Complex_Analysis_II/7.01%3A_Cauchy's_Theorem
    ∫zdz=∈2π0eitieitdt=i∫2π0ei2tdt=i∫2π0cos(2t)+isin(2t)dt=0 \[\begin{align*} \int f(z) dz &=\int u(x,y)+iv(x,y) dx+i \int u(x...∫zdz=∈2π0eitieitdt=i∫2π0ei2tdt=i∫2π0cos(2t)+isin(2t)dt=0 ∫f(z)dz=∫u(x,y)+iv(x,y)dx+i∫u(x,y)+iv(x,y)dy=∫u(x,y)dx−∫v(x,y)dy+i∫v(x,y)dx+i∫u(x,y)dy=∫bau(x(t),y(t))x′(t)−v(x(t),y(t))y′(t)dt+i∫bau(x(t),y(t))y′(t)+v(x(t),y(t))x′(t)dt

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