I've recently spent some time finding the cheapest fares for my journey and that got me thinking.  Essentially I'm traveling UK &lt;-> Tallinn (Estonia) anytime within the next 4 months for a week. But I'm not bothered travelling through other airports or countries if that makes my journey cheaper (and as a EU citizen with no checked in luggage all the transfers shouldn't be a big hassle).  So currently if I search for MAN &lt;-> TLL in flights.google.com that gives me the cheapest return for the end of May for about £190 with transfers (that's only because I've already bought my tickets so using end of May as an example to compare prices).  I live near Manchester but if I buy my tickets in advance (which I usually try to do) it doesn't matter if I fly from Manchester Airport, Edinburgh Airport, any airport in London or possibly anywhere from in the UK. Although it's less hassle to get back home from Manchester Airport - any airport in the UK presumably is fine for me as long as it's cheap (just an example to prove what I mean: I've spent £16.50 for the train ticket to get from Lancashire to Stansted Airport (London) and £14.05 to get from Manchester Airport back home, so there really is no big difference in price).  And if I choose London (all airports) &lt;-> TLL that gives me around £100 for a direct return.  So currently I have purchased the following tickets: STN->TLL (£36.71), TLL->DUB (~£23.84), DUB->MAN (~£7.94). So that's about £69 in total for a "return". And I could have reduced that price even more (by buying STN->RYG (£13) &amp; RYG->TLL (£12) instead of a direct one but I got a direct one because that one was better for me for that specific day only), or also had an option to fly MAN->DUB, DUB->TLL that would be again cheaper than my direct flight STN->TLL.  So the question is: I can of course find these routes and prices manually but it will take me days to look up every possible route between the UK and Tallinn (Estonia) and compare all the possible prices to narrow it down to 1-5 cheap options to choose from. I've used flights.google.com and skyscanner.net (and possibly some other ones) but none of them are able to find these cheap rotes that I can find manually (which is disappointing). Is there any website-software-anything that can find the cheapest tickets with layovers/transfers through other airports/countries from a country that I choose (for example the UK) to a country/city/airport (for example Estonia or Tallinn or TLL)?  P.S. Sorry for the long question but I had to provide my example to make it clear that the usual websites doesn't seem to find what I want to find.  EDIT1: Although broad searching is one aspect of my question, the most important one is actually finding the cheap connections automatically. Google or Skyscanner didn't see the possibility of flying cheaper like MAN->DUB->TLL or STN->RYG->TLL,  the only cheap option it knows about is the direct flight STN->TLL with Ryanair, although somehow the more expensive flights that it finds are with connections, so it can find flights with connections just not the cheap ones!  EDIT2: I've accepted Tor-Einar Jarnbjo's answer that explains why I couldn't find these cheap flights but if anyone knows a website that can still find these cheap flights please let me know! I accept all the risks!  EDIT3: As it was found out this question is a duplicate of  , although that original question was only answered because I've asked mine. :)
There are often situations where buying two separate one-way tickets from A to C via B is cheaper than buying a normal multi-leg ticket. This can be especially useful with airlines such as Ryanair, which specifically avoids selling multi-leg tickets.  Is there a search engine which allows one to search for such flights?
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Integrate $\int_{0}^{2\pi}\log|1-e^{i\theta}| d\theta$ Here is exercise 11, chapter 3 from Stein &amp; Shakarchi's Complex Analysis II:  Show that if $|a|&lt;1$, then:  $$\int_{0}^{2\pi}\log|1-ae^{i\theta}|\,d\theta = 0$$  Then, prove that the above result remains true if we assume only that $|a|\leq 1.$ ($a\in \mathbb{C}$)  I've already proved it for $|a|&lt;1$, but couldn't make it for $|a|=1$, which I thought would be the easy part. But the only thing I could come up with was that, WLOG, it is enough to prove that:  $$\int_{0}^{2\pi}\log|1-e^{i\theta}|\,d\theta = 0$$  Tried to look for some kind of symmetry, but could't find it. Any ideas? Thanks! Does a four-divergence extra term in a Lagrangian density matter to the field equations? Greiner in his book "Field Quantization" page 173, eq.(7.11) did this calculation:     ${\mathcal L}^\prime=-\frac{1}{2}\partial_\mu A_\nu\partial^\mu A^\nu+\frac{1}{2}\partial_\mu A_\nu\partial^\nu A^\mu-\frac{1}{2}\partial_\mu A^\mu\partial_\nu A^\nu $   $\space\space\space\space=-\frac{1}{2}\partial_\mu A_\nu\partial^\mu A^\nu+\frac{1}{2}\partial_\mu[A_\nu(\partial^\nu A^\mu)-(\partial_\nu A^\nu) A^\mu]$        The last term is a four-divergence which has no influence on the field equations. Thus the dynamics of the electromagnetic field (in the Lorentz gauge) can be described by the simple Lagrangian        ${\mathcal L}^{\prime\prime}=-\frac{1}{2}\partial_\mu A_\nu\partial^\mu A^\nu$     Yes, if it is a four-divergence of a vector whose 0-component doesn't contain time derivatives of the field, indeed according to the variational principle this four-divergence will not influence the field equation.  And actually I calculated the time-derivative dependence of 0-component of $[A_\nu(\partial^\nu A^\mu)-(\partial_\nu A^\nu) A^\mu]$, in which only $[A_0(\partial^0 A^0)-(\partial_0 A^0) A^0]$ could possibly contain time-derivative, which vanishes fortunately, so whatever the general case it doesn't matter in this present case.  But how can he seem to claim that it holds for a general four-divergence term,The last term is a four-divergence which has no influence on the field equations?  EDIT: I only assumed the boundary condition to be $A^\mu=0$ at spatial infinity, not at time infinity. 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