490 BELL SYSTEM TECHNICAL JOURNAL 



The roots^ of sin iaq = are 



flir 

 q„ = -:—, «=±1,±2, etc. 



Substituting these values of qn in above we get 



nirh 



1 +erf 

 , mr 

 - cosh -T- 



etc, 1 



. , J , sinh . 



sinh bq ^ » _^ ^ ^« g-c^^^^/a.,, 



sinhao a n=±i nir , mr 

 n=:±2 — ^cosh-T- 



If'-)] 



(-l)"sin-— 



___[_:£ y^ ^ g-(n27r2/a2)<. 



« 7r„=i. 2. 3. •.. W 



If (sinh bp'^'^)/(slnh ap^'^) is solved by the expansion theorem and the 

 summation is extended over both positive and negative roots, the result 

 is 



. mrb 



---{-- 2Z (~1)" ^ g-(.nwlai)t^ 



O- TT „=i, 2. 3, • • • ^ 



In other words the summation quantity is just double what it should be. 

 In order to correct this in practice, those who have used the theorem 

 for such cases have extended the summation only over the positive 

 roots, notwithstanding the fact that in similar cases with integral 

 exponents such as, for example, 1/cosh ap the summation is extended 

 over all the roots. The truth is the original expansion theorem is not 

 applicable if either numerator or denominator contains ^ to a fractional 

 form. In the above case were the problems to evaluate (sinh bp^'^/ 

 sinh ap"^'^) the expansion theorem gives an entirely incorrect answer, 

 while the correct answer is obtained from the extension to the theorem. 



Example 5. 



Here 



lA(/, qn) = e'"'' 



,1/3 



_ Q s- ?« 



i:7^Mt,qn). 



p2/3 _ 1 g2 _ 1 ^ 25„2 



gn = ± 1, 



1 Excluding the root g„ = which is not used in this case. 



