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Unformatted text preview: The Laplace Transform: Introduction; Definition Introduction: The Concept of a Transform 1. Logarithmic Representation: a Numerical Transform Let R + denote the set of positive real numbers and let b be a fixed positive number greater than 1, which we will call the base . For any r R + we can form the logarithm to base b of r ; we denote it by log b r . It is the unique real number with the property that b log b r = r. The most common bases are 10 and e = 2 . 1828 ... , leading to the common and natural logarithms, respectively. As we allow r to range through all of R + the values of log b r range over the whole set, R , of real numbers. Each logarithm, = log b r , thus constructed is an image , or transform of the original positive number r formed in a particular manner. The passage from back to r itself is just the defining identity displayed above, or r = b ; this can be thought of as the inverse transform corresponding to the logarithm transform. The logarithms log b r are more convenient than the original numbers r in certain ways, notably with respect to multiplica- tion, formation of powers, extraction of roots and differentiation of certain functions. Since log b ( rs ) log b r + log b s , we can form the product rs , which involves the relatively complicated operation of multiplication, by applying the simpler operation of addition to the logarithms of r and s . Equally well, we can obtain the power r p by multiplying log b r by p and then applying the inverse transform, i.e., taking the antilogarithm, to get r p itself. Thus the logarithm transform is useful precisely because cer- tain relatively complicated operations on positive numbers r, s , etc., carry over into simpler operations on the logarithms of those numbers....
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This note was uploaded on 01/23/2012 for the course MATH 4254 taught by Professor Robinson during the Spring '10 term at Virginia Tech.
- Spring '10
- Real Numbers