Theory of Differential Equations volume 4

Cover Theory of Differential Equations volume 4
Theory of Differential Equations volume 4
Forsyth, Andrew Russell, 1858-1942
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Hence the quantities Te", T^, ... , T^e" all remain finite within that range, that is, no one of them can become infinite, for a value of X sufficiently large* to make the quadratic form positive.
Next, suppose that the constants are complex, so that T, T lt ... Can have complex values ; but let t still be real. Then we write * For the tests, see Williamson s Differential Calculus, 3rd ed. , p. 408. In the case of k = 4, the conditions are : \>0, so that it is sufficient to take X greater than
...the greatest positive value which makes the left-hand side in the last inequality vanish.
102. ] THEOREM 321 for all values of r, where (f> r and ^ r are real ; the system of equations takes the form -F- -**~ *&-, _ where > =0, ^r Q = ^, and P s =p s + iq s . We now have 2A; equations; on substituting they give the modified set r - 1 = - (Pi - Hence r=l fc-2 :X 2 I k-l , 2 + ^ r 2 ) + (X - i + ) + bilinear terms.
As before, by choosing a sufficiently large value of X, the right- hand side can be made always positive.


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