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In Spain's book, Functions of mathematical physics he introduces the contour integral method of solving ODEs. The baseic idea is:

given an ODE $\sum_0^m a_r(t) \frac {d^rf}{dt^r} = 0$, a solution may be sought after in the form:

$$ \int_c{k(z,t)v(t)dt} $$

The ingenuity is really in the choice of $k$ and a contour can then be chosen to make sure 'boundary terms' vanish. The apeal is of course for example in finding integral representations of special functions. The laplace method is when choice of $k$ is of the form $k = e^{t*h(z)}$ (in general) and we frequently end up with a mellin inversion formula or similar looking solution after finding the solution to a much simpler problem satisfied by $v(t)$. For example the integral representation of airy functions.

Another popular choice of $k$ is of the form $ k = (t - z)^\mu $ which is called the euler transformation. where $\mu$ is a parameter to be determined. For example, the integral representation of gauss's hypergeometric function looks like this.

for example the nth order ODE with linear polynomial coefficients:

$$ \sum_{k=0}^n {(a_k + b_kt)\frac {d^kw}{dt^k}} = 0; $$

We can try a laplace-type choice for $k$ say $k = e^{zt}$ which upon substitution yields two polynomials in z:

$F(z) = \sum_{k=0}^n{a_kz^k}$ and $G(z) = \sum_{k=0}^n{b_kz^k}$ and the ODE is tranformed to:

$$ \int_c {e^{zt} v(z)(F(z)+tG(z))}dz = 0 $$

Upon integration by parts (i'll ommit the z's, primes denote differntiation wrt z):

$$\int_c{e^{zt}(Fv - (Gv)^\prime)dz} + [GSe^{zt}]_{C} = 0$$

We then choose a contour that ensures the last term in the LHS above vanishes and solve the 1st order system in the parentheses in $z$ intead to find $v$ as:

$$v(z) = \frac {A}{G(z)} exp\{\int^z \frac {F(z)}{G(z)} dz\}$$

for some arbitrary constant.

In general tis type of choice of $k$ 'reflects the order' of the problem to be solved. For example above an nth order ODE is solved by finding a 1st order ODE that $v$ satisfies. If the order of the polynomials in $a_k,b_k$ were quadratic we would have a 2nd order ODE that $v$ satisfies thereby not have simplified the problem a great deal. This is usually when the euler-type choice of k is useful. For example in proving the integral representation of legendre polynomials.

Sorry for the long preamble, Its to get others interested in the elegance of this as I see it isnt very popular.

My question is: Is there a more modern text that covers this method of solving ODEs? Perhaps the name 'euler transformation' is not the most popular name for the second choice of $k$, if so does anyone know another name for the method? Alternatively does anyone know a different proof of the integral representation of legrendre polynomials and how to show equivalence of the different types?

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    $\begingroup$ would you be more specific?! $\endgroup$ Sep 4, 2015 at 18:55

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