F(z) = e^z/z

Definition

The function f(z) = e^z/z is a complex function defined for complex numbers, where e^z represents the exponential function and z is in the denominator. This function has interesting properties, including essential singularity at z = 0, and its behavior can be analyzed using the residue theorem for integration over closed contours. This function also provides insights into the nature of singularities in complex analysis.

5 Must Know Facts For Your Next Test

1. The function f(z) = e^z/z has an essential singularity at z = 0, which means it does not have a limit as z approaches 0.
2. To find the residue at z = 0 for f(z), one can use the series expansion of e^z, which simplifies the calculation significantly.
3. The residue theorem states that the integral of f(z) around a closed contour is 2πi times the sum of residues inside that contour.
4. When integrating f(z) over a closed contour that encircles z = 0, one can easily compute the integral using its residue at that point.
5. The behavior of f(z) as z approaches infinity can be examined to show that it approaches 0, which is useful in various applications of complex analysis.

Review Questions

• How does the presence of an essential singularity at z = 0 affect the behavior of the function f(z) = e^z/z?
  • The essential singularity at z = 0 means that as z approaches 0, the behavior of f(z) becomes increasingly unpredictable and does not converge to any particular value. Instead, it exhibits oscillatory behavior. This impacts not just the evaluation of limits but also how we approach integration around this point since traditional methods may not apply directly.
• What steps would you take to calculate the residue of f(z) = e^z/z at z = 0, and why is this significant for applying the residue theorem?
  • To calculate the residue at z = 0, first expand e^z into its Taylor series: $$e^z = 1 + z + \frac{z^2}{2!} + \cdots$$. Dividing this series by z gives us the Laurent series from which we can identify that the residue is 1 (the coefficient of (z - 0)^{-1}). This is significant because once we have determined this residue, we can apply the residue theorem to compute integrals involving f(z) around contours enclosing z = 0.
• Discuss how analyzing f(z) = e^z/z contributes to our understanding of complex functions and their singularities in general.
  • Analyzing f(z) = e^z/z allows us to see how essential singularities behave compared to poles or removable singularities. Understanding these differences helps us grasp more complex functions' behavior in various applications. By studying this function and employing tools like the residue theorem, we learn about contour integration's power in simplifying seemingly complex problems and gaining deeper insights into holomorphic functions and their singularities.

Related terms

Residue: The residue of a function at a singular point is the coefficient of (z - z₀)^{-1} in its Laurent series expansion around that point, crucial for evaluating integrals.

Laurent Series: A representation of a complex function as a series that includes terms of negative powers, which allows analysis around singular points.

Contour Integration: A method of evaluating integrals along paths (contours) in the complex plane, often used in conjunction with the residue theorem.