Okay, thanks, what was not so clear to me is how to use the "three dots"
notation to hand some points over to the integration function. For example,
to integrate 1/z along the unit circle around the pole at the origin:
julia> x = linspace(0, 2*pi);
julia> points = exp(x*1im);
julia> quadgk(z -> 1 ./ z, points...)
(-2.1630337481358435e-17 + 6.283185307179583im,5.957695483631124e-16)
That is the correct solution 2 \pi i to a very good accuracy.
On Friday, July 18, 2014 6:53:58 PM UTC+2, Steven G. Johnson wrote:
>
>
>
> On Friday, July 18, 2014 8:06:25 AM UTC-4, Hans W Borchers wrote:
>>
>> First, I think, Steven is talking about line integrals along a path, and
>> these integrals are one-dimensional.
>>
>> And secondly, of course you should avoid poles along your path, by making
>> the radius big enough, or similar. These things are standard techniques in
>> complex analysis.
>>
>
> Note also that if you have poles "on" your path, then you need more
> information in order to define what your integral even means. e.g. do you
> mean the limit as the pole approaches the path (or vice versa), and if so
> you need to specify from what side you are taking the limit. Then you can
> (for example) deform your integration contour away from the pole (in the
> correct direction) as Hans suggests.
>
> As an example, I append a function `line_integral` from my package
>> NumericalMath (not in METADATA) that does integrate along a path given by
>> `points`.
>>
>
> Hans, note that quadgk already supports line integration over straight
> line-segments in the complex plane along any arbitrary sequence
> points::Array{Complex} of points. Just do:
>
> quadgk(f, points...)
>
> This is better than doing the integral over each line segment separately,
> because it is globally adaptive: in order to decide whether it needs to
> subdivide any interval to improve accuracy, it will look at *all* the
> intervals, not just the intervals within a single line segment.
>
> Also, there is no need to integrate the real and imaginary parts
> separately. quadgk can integrate complex-valued functions (and
> vector-valued functions, and matrix-valued functions, and...) just fine.
> By default, it measures the error in the usual norm sqrt(real^2 + imag^2),
> but if you want to use min(|real|,|imag|) instead (which is effectively
> what you are doing if you integrate them separately), then you can pass
> that norm for the "norm" keyword argument.
>
> --SGJ
>
