text: # Integral of e^(-z) dz / (z - (pi*i)/2)
  
  1. Apr 20, 2007
  
  ### laura_a
  
  1. The problem statement, all variables and given/known data
  
  Let C denote the positively oriented boundary of the square whos sides lie on the lines (x= + / - 2 and y = + / - 2)
  
  Evaluate int_c (e^(-z) dz) / (z - (pi*i/2))
  
  2. Relevant equations
  
  This is a text book question after the topic of Cauchy Integrals but before residues
  
  3. The attempt at a solution
  
  I am not very cluely at the Contour integration as the teacher just gave us the text book notes and said do it... so I figure to use the Cauchy Integral
  
  f(z_0) = 1/(2*pi*i) int_c f(z) dz / z - z_0
  
  Where my belief is that z_0 is interior to the given contour (in this case it's the square)
  
  so I thought that the question looks like it will just fit inside the formula, so to speak, because (pi*i)/2 is interor to the contour (well if it's not then I have no idea what i'm talking about) so then i put
  
  f(z_0) = 1/(2*pi*i) int_c e^z dz/ z - ((Pi*i) / 2)
  
  so I sub in f(z_0) = e^(Pi*i) and get -1
  
  so -1 = 1/(2*pi*i) int_c e^z dz/ z - ((Pi*i) / 2)
  -2*pi*i = int_c e^z dz/ z - ((Pi*i) / 2)
  
  and thats where I stop :surprised because I dont know how to integrate e^z dz/ z - ((Pi*i) / 2)
  
  Any suggestions?
  
  2. Apr 20, 2007
  
  ### Päällikkö
  
  Cauchy's integral formula (I'm not gonna list the assumptions it requires, you should make sure that they are satisfied in this problem, though):
  $$f(\alpha) = \frac{1}{2\pi i}\int_C \frac{f(z)}{z-\alpha}dz$$
  
  Substitute f(z) = e-z:
  
  $$\int_C \frac{e^{-z}}{z-\alpha}dz = 2\pi i e^{-\alpha}$$
  
  Can you do the rest?
  As you can see, you don't really have to integrate anything.
  
  Last edited: Apr 20, 2007
metadata: {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}