I found this example in a textbook dated 1937 which I use as supplementary material for M828. It gave me some hard time but finally sorted out many fine tricks of contour integration. Some inspiration was provided by the discussion of the Pochhammer's extension of the Eulerian integral of the first kind by Whittaker and Watson.
Evaluate the following integral:
integral over zero under one root of order four over four divided by open one plus x close cubed d x
Consider the integral in the complex plane:
cap j equals contour integral over cap c root of order four over four divided by open one plus z close cubed d z
where contour C is constructed as follows. Make a cut along the segment
open zero comma one close
arg of z equals zero
z equals one
equation left hand side cap c sub gamma super one equals right hand side open z colon absolute value of one minus z equals gamma comma normal black letter cap i times z greater than zero close
three times pi divided by four
z equals cap r
equation left hand side cap c sub cap r equals right hand side open z colon absolute value of z equals cap r close
This circle will enclose both branch point of the integrand
z equals zero
z equals one
equation sequence z super one divided by four times e super two times pi times i times one divided by four times open one minus z close super three divided by four times e super two times pi times i times three divided by four equals z super one divided by four times open one minus z close super three divided by four times e super two times pi times i equals z super one divided by four times open one minus z close super three divided by four
This is the reason why we only need to make the cut along the segment
open zero comma one close
z equals cap r
equation left hand side cap c sub gamma squared equals right hand side open z colon absolute value of one minus z equals gamma comma normal black letter cap i times z less than zero close
z equals one
three times pi divided by four
equation left hand side cap c sub delta equals right hand side open z colon absolute value of z equals delta close
equation left hand side two times pi times one divided by four equals right hand side pi divided by two
As the result of this construction the integral is split into the following parts:
multiline equation row 1 cap j equals sum with, 4 , summands integral over delta under one minus gamma f of z d z plus integral over cap c sub gamma super one f of z d z plus e super negative three times pi times i divided by four times integral over one plus gamma under cap r f of z d z plus integral over cap c sub cap r f of z d z plus row 2 sum with, 4 , summands prefix plus of e super five times pi times i divided by four times integral over cap r under one plus gamma f of z d z plus integral over cap c sub gamma squared f of z d z plus e super pi times i divided by two times integral over one minus gamma under delta f of z d z plus integral over cap c sub delta f of z d z
Two integrals around the small circles add up to an integral over a circle:
equation sequence integral over cap c sub gamma super one f of z d z plus integral over cap c sub gamma squared f of z d z equals integral over cap c sub gamma f of z d z equals integral over zero under two times pi root of order four over four divided by open two minus gamma times e super i times theta close cubed times i times gamma times e super i times theta d theta
lim over gamma right arrow zero of integral over cap c sub gamma f of z d z equals zero
Similarly, the integral around a small circle around the origin vanishes:
equation left hand side integral over cap c sub delta f of z d z equals right hand side integral over zero under two times pi root of order four over four divided by open one plus delta times e super i times theta close cubed times i times delta times e super i times theta d theta
lim over delta right arrow zero of integral over cap c sub delta f of z d z equals zero
Integrals along the segment
open one plus gamma comma cap r close
multiline equation row 1 equation left hand side e super negative three times pi times i divided by four times integral over one plus gamma under cap r f of z d z plus e super five times pi times i divided by four times integral over cap r under one plus gamma f of z d z equals right hand side e super negative three times pi times i divided by four times integral over cap r under one plus gamma f of z d z plus e super two times pi minus three times pi times i divided by four times integral over cap r under one plus gamma f of z d z equals row 2 equation sequence equals e super negative three times pi times i divided by four times integral over cap r under one plus gamma f of z d z minus e super negative three times pi times i divided by four times integral over cap r under one plus gamma f of z d z equals zero
The integral over the large circle also tends to 0 as R increases. This can be shown using the Jordan's lemma, or by direct calculation:
equation sequence integral over cap c sub cap r f of z d z equals integral over zero under two times pi root of order four over four divided by open one plus cap r times e super theta close cubed times i times cap r times e super i times theta d theta equals integral over zero under two times pi root of order four over four divided by open one divided by cap r plus e super i times theta close cubed times i times e super i times theta divided by cap r d theta
lim over cap r right arrow normal infinity of integral over cap c sub cap r f of z d z equals zero
Finally the only two terms that survive allow us to express the contour integral in terms of the integral along the segment of the real axis:
multiline equation row 1 equation sequence cap j equals integral over zero under one f of z d z minus e super pi times i divided by two times integral over zero under one f of z d z equals open one minus e super pi times i divided by two close times integral over zero under one f of z d z equals e super pi times i divided by four times open e super negative pi times i divided by four minus e super pi times i divided by four close times integral over zero under one f of z d z equals row 2 equation left hand side equals right hand side negative two times i times e super pi times i divided by four times sine of pi divided by four times integral over zero under one f of z d z
The contour
cap c
z equals negative one
cap j equals two times pi times i times cap r sub negative one times f of z
equation left hand side integral over zero under one f of z d z equals right hand side negative Square root of two times pi times e super negative pi times i divided by four times cap r sub negative one times f of z
The residue can be calculated using the standard formula:
equation sequence cap r sub negative one times f of z equals one divided by two times lim over z right arrow negative one of open open one plus z close cubed times root of order four over four divided by open one plus z close cubed close super double prime equals one divided by two times lim over z right arrow negative one of open root of order four over four close super double prime
Calculation of the derivative can be facilitated by taking logarithm first:
g equals root of order four over four
natural log of g equals one divided by four times natural log of z plus three divided by four times natural log of one minus z
equation sequence g super prime divided by g equals one divided by four times z minus three divided by four times open one minus z close equals one minus four times z divided by four times z times open one minus z close
equation left hand side g super prime equals right hand side one minus four times z divided by four times z super three divided by four times open one minus z close super one divided by four
natural log of g super prime equals natural log of one divided by four plus natural log of one minus four times z minus three divided by four times natural log of z minus one divided by four times natural log of one minus z
equation sequence g super double prime divided by g super prime equals negative four divided by one minus four times z minus three divided by four times z plus one divided by four times open one minus z close equals negative three divided by four times open one minus four times z close times z times open one minus z close
equation left hand side g super double prime equals right hand side negative three divided by 16 times z super seven divided by four times open one minus z close super five divided by four
equation sequence cap r sub negative one times f of z equals one divided by two times lim over z right arrow negative one of negative three divided by 16 times z squared times open one minus z close times z super one divided by four divided by open one minus z close super one divided by four equals negative three times e super pi times i divided by four divided by 64 times root of order four over four
equation sequence integral over zero under one root of order four over four divided by open one plus x close cubed d x equals negative Square root of two times pi times e super negative pi times i divided by four of negative three times e super pi times i divided by four divided by 64 times root of order four over four equals three times pi times root of order four over four divided by 64
