$SPAD/src/input branchcut.input

Daniel Zwillinger December 3, 2016

Abstract

This example is from the Handbook of Integration, p 58

Contents

2 Consider the integral

I(a, b) = ∫_{0}^{1} \frac{dx}{ax + b}

for real x and arbitrary nonzero complex a and b. The indefinite integral has the primitive log(ax + b)/a. Hence, a careless “direct” derivation would yield the result

I(a, b) = \frac{log(a + b)}{a} − \frac{logb}{a}

The problem, of course, is that the logarithm function has a branch cut. Hence, the two logarithms may not be on the same Riemann sheet.

The correct way to evaluate it is to separate the region of integration into two subintervals, with the division point being the value where ax + b may vanish. An easier way, for this integral, is to first write the integral as

I(a, b) = \frac{1}{a} ∫_{0}^{1} \frac{dx}{x + \frac{b}{a}} = \frac{1}{a} (x + \frac{b}{a}) |_{0}^{1}

No matter what the sign of the Im(b/a), the argument of the logarithm never crosses the cut (since x is real). Thus, the answer is

I(a, b) = \frac{1}{a} [log (1 + \frac{b}{a}) − log \frac{b}{a}]

Note that since 1 + b/a and b/a have the same imaginary part, we may combine them to obtain our final answer

I(a, b) = \frac{1}{a} log \frac{a + b}{b}

— * —

)set break resume
)sys rm -f branchcut.output
)spool branchcut.output
)set message test on
)set message auto off
)clear all
--S 1 of 5
t1:=integrate(a/(a*x+b),x=0..1,"noPole")
--R
--R
--R             2           2         2
--R        log(b  + 2a b + a ) - log(b )
--R   (1)  -----------------------------
--R                      2
--R                  Type: Union(f1: OrderedCompletion(Expression(Integer)),...)
--E 1

--S 2 of 5
t2:=(1/a)*log((a+b)/b)
--R
--R
--R            b + a
--R        log(-----)
--R              b
--R   (2)  ----------
--R             a
--R                                                    Type: Expression(Integer)
--E 2
--S 3 of 5
complexNormalize t1-t2
--R
--R
--R               2           2           2         b + a
--R        a log(b  + 2a b + a ) - a log(b ) - 2log(-----)
--R                                                   b
--R   (3)  -----------------------------------------------
--R                               2a
--R                                                    Type: Expression(Integer)
--E 3
--S 4 of 5
t3:=log(a+b)/a-log(b)/a
--R
--R
--R        log(b + a) - log(b)
--R   (4)  -------------------
--R                 a
--R                                                    Type: Expression(Integer)
--E 4
--S 5 of 5
complexNormalize t1-t3
--R
--R
--R               2           2           2
--R        a log(b  + 2a b + a ) - a log(b ) - 2log(b + a) + 2log(b)
--R   (5)  ---------------------------------------------------------
--R                                    2a
--R                                                    Type: Expression(Integer)
--E 5
)spool
)lisp (bye)

———-

References^{[1] nothing}

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