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Cauchy's integral formula expresses \( f(a) \) for \( a \) inside \( C \) as

  • A.\( \dfrac{1}{2\pi}\displaystyle\oint_{C} \dfrac{f(z)}{z-a}\,dz \)
  • B.\( \dfrac{1}{2\pi i}\displaystyle\oint_{C} \dfrac{f(z)}{z-a}\,dz \)
  • C.\( \dfrac{1}{2\pi i}\displaystyle\oint_{C} f(z)(z-a)\,dz \)
  • D.\( 2\pi i \displaystyle\oint_{C} \dfrac{f(z)}{z-a}\,dz \)

Answer: B

The formula reads

\[ f(a) \;=\; \frac{1}{2\pi i}\oint_{C} \frac{f(z)}{z - a}\,dz, \]

so the values on the contour determine the value at every interior point. The factor \( 2\pi i \) belongs in the denominator, and the \( i \) with it: without the \( i \) the formula would not even be dimensionally consistent as a complex identity

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