The phrase "continuous variation" appears several times in Titchmarsh's books and is repeated verbatim by those who cite his work. But nowhere can I find this phrase defined. The earliest reference I have found (1939) occurs on page 132 of his book "The Theory of Functions". From the context it seems to describe a particular way of defining the value of a cut function. Can anyone provide a (citable) reference to a formal definition of this phrase, or, better yet, an explanation of its meaning?
In both, mathematics including statistics and biosciences, term continuous is frequently used.
In mathematics, especially in calculus and topology, a continuous function is, roughly speaking, a function for which sufficiently small changes in the input result in arbitrarily small changes in the output. Otherwise, a function is said to be a discontinuous function. A continuous function with a continuous inverse function is called a homeomorphism.
In biosciences, it mostly used in case of genetic variation and also in case of body weight and height (phenotypic traits).
If all members of a species have the same set of genes, how can there be genetic variation? As indicated earlier, the answer is that genes come in different forms called alleles. In a population, for any given gene there can be from one to many different alleles; however, because most organisms carry only one or two chromosome sets per cell, any individual organism can carry only one or two alleles per gene. The alleles of one gene will always be found in one chromosomal position. Allelic variation is the basis for hereditary variation.
http://www.biologymad.com/resources/RevisionM5Ch2.pdf
http://www.math.umn.edu/~olver/ln_/cv.pdf
http://math.stanford.edu/~katznel/restrictionsfeb25.pdf
https://msu.edu/course/plb/849/Conner/ConnerHartlCh4.pdf
http://onlinelibrary.wiley.com/doi/10.1111/j.1601-5223.1949.tb03348.x/pdf
https://www.whitman.edu/Documents/Academics/Mathematics/grady.pdf
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1213897/pdf/391.pdf
[Book] An Introduction to Genetic Analysis; 7th edition
Thank you both for trying. Obviously I did not pose the question clearly, so let me try again. In his (1939) book "The Theory of Functions", Titchmarsh writes:
"Let f(z) be analytic and not zero on C, and meromorphic inside it. Let F(z)=log(f(z)), the logarithm being defined as follows: we start with a particular determination on x=x2, and obtain the value at other points by continuous variation along y=constant from log(x2+iy)". What is the precise mathematical definition of "continuous variation" as it is used in this sentence? What does this sentence mean in itself? Thank you.
Links 2 and 3 and the book listed provides a clear answer thouh not in your terms of formulation of function.
A continuous function is that which can take any value according to function of x on y.
Getting back to the question: (Mohammad - Thank you, but I know the meaning of "continuous"). In his book, Titchmarsh is attempting to derive Littlewood's theorem, which, in essence, states that the contour integral of F(z)=log(f(z)) over a region equals the excess of the number of zeros over the number of poles of f(z) enclosed in that region. It is inconceivable that Titchmarsh would use the term "continuous variation" to prove a theorem unless that term has a precise meaning. I tend to agree with Peter's interpretation - that was my guess as well. But guessing isn't good enough when it comes to proving theorems. Somewhere that phrase must be precisely defined. Hence the question.
I had actually started at the book on the Riemann Function that you found, and worked my way back to the text I cited, but never found the references to Newton or anything earlier. I suppose that pretty much settles the question. Thank you.
@michael. The continuous variation TIchmarsch means is a so called homotopy of (closed) paths \gamma in the complement of the zero's and poles of the holomorphic function f(z). We can then define
d\log(f(z)) = (f'(z) / f(z) ) dz .
The theorem that Titschmarsch refers to is
\int_\gamm d\log(f(z)) = \int_\gamma (f'(z))/f(z) dz = 2\pi i (#zero's - #poles) encircled by \gamma.
Now let D \subset C be the complement of the zero's and poles. A closed path is just a continuous map of the interval
\gamma: [0,1] \to D
such that \gamma(0) = \gamma(1) . A continuous variation of gamma is what is called a homotopy, Two closed path's \gamma_0 and \gamma_1 are homotopic if there is continuous map
\Gamma : [0,1] \times [0,1] \to D
such that
\Gamma(t, 0) = \gamma_0(t); \Gamma(t, 1) = \gamma_1(t)
and
\Gamma(0, s) = \Gamma(1,s)
i.e. if write \gamma_s(t) = \Gamma(t,s), then the family of paths \gamma_s continuously deform the path \gamma_0 into \gamma_1 (all the time avoiding the zero's and poles of f!). Now what Titchmarse says is that if \gamma_0 and \gamma_1 are homotopic then
\int_{\gamma_0} d\log (f(z)) = \int_{\gamma_1} d\log (f(z))
i.e. the integral only depends on the homotopy class of the path.
Remark: it so happens that the integral even only depends on the homology class of gamma. This is a bit harder to explain but here it means that as far as contour integration is concerned, if two (possibly disconnected) contours bound a surface not meeting the singularities they are equivalent. In practice this allows you to reduce a contour integration of a meromorphic function to integrating over small circles encircling the singularities (which for a function of the form f'(z)/f(z) are the zero's and poles of f). This a modern formulation of a classical fact and may well be what Titchmarsch actually uses.
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