@vv 

Thanks

A spacial complex function what rises out from the complex plane, like the riemann zeta function is, cannot sterographical projected on the riemann sphere.

The projection must be done from the flat complex plane, right?

The complex function as mountain landscape what possible posesses all sorts of zeroes points and sorts of poles what is visible.
More difficult is to do calculation on this. 
The order of  a zero is highest needed derative to get it 0 , or  the pole calculation is with a limit process and with a residu (must study it further, is not complete correct what i write i think) 
Much more not intuiative is intergrating, with a contour and a loops and don't have for this not a general approach yet, especilaly for the loops , what all is involved?

@mmcdara 

Thanks 
Thought it was a help for visualizing complex functions on the Riemann sphere
Probably not used by Riemann himself.

Cartesian to spherical coordinates transformation differs from a  stereographic projection?

@Scot Gould 
Thanks

This a better presentation than in maple input format , that's my impression.
Also the use of a prime makes that the differential equation is easier for input handling and reading.

Is there a idea what this function y(x) is notated? 

@ecterrab 

Thanks

Enough information and helpful in order for me to make a further start with this Zeta function.

@dharr 
Thanks

I will study it further.

@tomleslie 

Thanks

Looks great this plot and how it is made

The non-trivial zeroes of the Zeta function are all positioned on this critical line 
Knowing a formula for this zeroes positions occur on the 1/2 line, means that you have solved the Riemann Hypothese ( you win a prize ..bingo )
I wonder if there will be ever found a formula what predicts the distribution of primes ?

There is now a calculation going on for years now to find the next mersenne prime

So my question was not possible realizing this afterwards. 
I do want also construct later a 3 d complex plot from the critical line (strip)

@vv 

Thanks

Yes, the manual calculation shows your answer too 
A lot of work to do by hand.
Certain classes of real integrals can be solved with complex integration.

@Carl Love 

Thanks

"you must use option continuous or the integral simply won't be done. It's as simple as that, it's nothing particular to this problem, and I don't understand why it causes you confusion."

In the past it could be done without  this option continuous adding now ?..that's the cause of my confussion
But it has to to do with the variable t used here then , as i understand it now correctly from your explanation .

I got two answers for the Differential equation: i was curious if they are the same ( in fact a needless thing to ask myself, because Maple is error proof here)

Its now clear that i writing directly here 

@janhardo 

I figured out that this integral

  probably can be solved by residu calculus (pole and singularity)
I am wondering further what kind of integrals cannot be solved by Maple and complex calculus is needed ? 

@dharr 

Thanks

Indeed looking now closely to integral outcome : its a number : y= 1.155....
Why the tutor  replaces the original real integral with a complex one, what looks similar?

evalf(Pi*cosh(1) - Pi*sinh(1));
                          1.155727349

@Carl Love 

Thanks

Seems that the seq command not symbolic sums can do , only numeric 
Wondering why it is not possible with the sum command to show some terms in de serie ( this is student functionality )

@janhardo 
How to plot the graph of this function ?

Download youtube_vb_uitleg_berekening_reel_integraal_vi_acontour_integraal.mw

@vv 

Thanks

Forget this edward's book you said for learning complex analysis, that's a good advise.
There is much study material to find for learning complex analysis

It's only that i do some investigations with Maple about complex analysis how define it.

There is another modern book about the Riemann Zeta from Alexsandar ivic

Don't worry, i  am learning for the most part  not complex analysis from the book of Edwards.  

@vv 

Thanks

Yes, i am real beginner In this Complex Analysis.

Got interested in the Riemann hypothesis and wondered how he got his results.
Trying to track that with Maple seemed like a nice challenge.
The complex integral you asked about is in the book: riemann's zeta function : h.m .edwards  
P.S if you are interested in the book?

This integral is used to derive the functional formula of the Zeta(s) function as it seems.

@Kitonum 

Thanks

With this from you , every  partiele sum can then be written