Re: Calculating Sums Of Roots For Trans. Functions

• To: mathgroup at smc.vnet.net
• Subject: [mg5536] Re: [mg5498] Calculating Sums Of Roots For Trans. Functions
• From: Allan Hayes <hay at haystack.demon.co.uk>
• Date: Thu, 19 Dec 1996 01:02:33 -0500
• Sender: owner-wri-mathgroup at wolfram.com

```frankst at deakin.edu.au (Frank Stagnitti)
[mg5498] Calculating Sums Of Roots For Trans. Functions
writes

>>>>>>>>>>>
How do i calculate the first n roots of tan(z) = -z, and then sum
them in one command.

I think i need the right combinations of Findroot, Table and N/Sum
commands.

Findroot and Table give me a non-unique and non-ordered set of roots e.g.

RootTable = Table[{FindRoot[Tan[z]==-z, {z,i},
MaxIterations ->50],i},{i,0,EndRange,Pi}]
// TableForm

In short, i'd like a procedure/command that essentially sums the first n
(unique and ordered) roots
of tan(z) = - z with no user intervention

Any ideas??

<<<<<<<<<<<<<<

Frank,

From
Plot[{-x, Tan[x]}, {x,0,20}]
and a little experimentation we get the sum of the first fifty
positive roots as follows
Sum[ z/.FindRoot[Tan[z] == -z, {z,Pi/2 + n Pi + .02/(n+1)},
MaxIterations ->50
],
{n,0,49}
]

3928.67

The erratic behaviour shown by your code:

RootTable = Table[{FindRoot[Tan[z]==-z, {z,i},
MaxIterations ->50],i},{i,0,60,Pi}]// TableForm

z -> 0.        0
z -> 2.02876   Pi
z -> 2.02876   2 Pi
z -> 2.02876   3 Pi
z -> 2.02876   4 Pi
z -> 7.97867   5 Pi
z -> 2.02876   6 Pi
z -> 2.02876   7 Pi
z -> 2.02876   8 Pi
z -> 14.2074   9 Pi
z -> 7.97867   10 Pi
z -> 4.91318   11 Pi
z -> 2.02876   12 Pi
z -> 20.4692   13 Pi
z -> 2.02876   14 Pi
z -> 20.4692   15 Pi
z -> 2.02876   16 Pi
z -> 26.7409   17 Pi
z -> 14.2074   18 Pi
z -> 23.6043   19 Pi

is explained by

Plot[{-x, Tan[x]}, {x,0,20}]

Try drawing a tangent to the curve at the points with these x
coordinates, where it meets the x-axis is the first step in the root
finding process.

We can track what happens starting at 12Pi as follows:

FindRoot[Tan[Print[z];z] == -z, {z,12Pi}]
z
37.6991
18.8496
9.42478
4.71239
7.06858
4.37906
3.6763
1.86049
1.97323
2.02324
2.02871
2.02876
2.02876
{z -> 2.02876}

Allan Hayes
hay at haystack.demon.co.uk
http://www,haystack.demon.co.uk

```

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