Examples of the Processes of the Differential and Integral Calculus |
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Page 4
... roots and powers , it is generally most convenient to take the differential of the logarithm , or , as it is usually called , the logarithmic differential of the function . ( 31 ) Let u = ( a + x ) " ( b + x ) " , ( 32 ) ( 33 ) ( 34 ) ...
... roots and powers , it is generally most convenient to take the differential of the logarithm , or , as it is usually called , the logarithmic differential of the function . ( 31 ) Let u = ( a + x ) " ( b + x ) " , ( 32 ) ( 33 ) ( 34 ) ...
Page 61
... root of this is 2 , and if we take it , we find by the same method as in the last example the series 1 X03 4 u = 2 + x - + 2 1.2.3 20 1.2.3.4 + & c . The other series for u would be found by taking the impossible values of the cube root ...
... root of this is 2 , and if we take it , we find by the same method as in the last example the series 1 X03 4 u = 2 + x - + 2 1.2.3 20 1.2.3.4 + & c . The other series for u would be found by taking the impossible values of the cube root ...
Page 65
... root of the equation . ( 6 ) Let y3- ay + b = 0 : expand y " in terms of 1 a Here f ( x ) = x ” , $ ( x ) = x3 , b ≈ = - Whence br y " = 1 + n a2 a b2 1 n ( n + 5 ) b ' 1 n ( n + 9 ) ( n + 10 ) ( n + 11 ) 1 2.3.4 n ( n + 7 ) ( n +8 ) 6 1 + ...
... root of the equation . ( 6 ) Let y3- ay + b = 0 : expand y " in terms of 1 a Here f ( x ) = x ” , $ ( x ) = x3 , b ≈ = - Whence br y " = 1 + n a2 a b2 1 n ( n + 5 ) b ' 1 n ( n + 9 ) ( n + 10 ) ( n + 11 ) 1 2.3.4 n ( n + 7 ) ( n +8 ) 6 1 + ...
Page 66
... root . ( 9 ) If the equation be cy2 - by + a = 0 , of which the two roots are a , ẞ , then - 1.2 1 1 ne a n ( n { 1 + B b b 5 ) c ( a ( ) n ( n = 4 ) ( n -- 1.2 . 3 b b 3 3 ) c + & c . } , ( the series only continuing so long as there ...
... root . ( 9 ) If the equation be cy2 - by + a = 0 , of which the two roots are a , ẞ , then - 1.2 1 1 ne a n ( n { 1 + B b b 5 ) c ( a ( ) n ( n = 4 ) ( n -- 1.2 . 3 b b 3 3 ) c + & c . } , ( the series only continuing so long as there ...
Page 67
... roots of the transformed equation , we obtain a series for the direct th powers of the roots of the original equation . ( 11 ) If we thus transform the equation in Ex . 10 , it becomes c - by + ay2 = 0 ; and if a , ẞ be the same ...
... roots of the transformed equation , we obtain a series for the direct th powers of the roots of the original equation . ( 11 ) If we thus transform the equation in Ex . 10 , it becomes c - by + ay2 = 0 ; and if a , ẞ be the same ...
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