combine arctangent terms - Maple Help

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combine/arctan - combine arctangent terms

Calling Sequence

combine(f, arctan)

combine(f, arctan, m)

Parameters

f

-

any expression

m

-

the name 'symbolic'

Description

• 

The command combine(f, arctan) combines sums of arctangents in expressions by applying the following transformations:

arctanx+arctany=csgnx2+1xπ2x=1yarctanx+yxy+1xy<1orcsgnxcsgnyarctanx+yxy+1+csgnxπotherwise

2

  

If the input is a difference of two arctangents arctan(x) - arctan(y) then the above transformations are applied to arctan(x) + arctan(-y) .

• 

If the conditions required for the transformations cannot be determined by Maple, then the arctangents are not combined.  If the optional argument symbolic is specified, and the conditions cannot be determined, then transformation (2) is applied ``regardless''.

• 

Note, that in order to determine whether the transformations rules can be applied, one must be able to write an expression in the form

  

a&plus;barctanc±arctand .

  

This is not always easy to do so the code may fail to combine arctangent terms because of this.

Examples

f:=arctan1I&plus;arctan12&plus;1I2

f:=arctan1I&plus;arctan12&plus;12I

(1)

combinef&comma;arctan

12&pi;

(2)

f:=arctan13&plus;arctan15&plus;arctan17&plus;arctan18

f:=arctan13&plus;arctan15&plus;arctan17&plus;arctan18

(3)

combinef&comma;arctan

14&pi;

(4)

f:=arctan13&plus;arctan14

f:=arctan13&plus;arctan14

(5)

combinef&comma;arctan

arctan711

(6)

f:=3arctan132arctan14&plus;arctan15

f:=3arctan132arctan14&plus;arctan15

(7)

combinef&comma;arctan

arctan427536

(8)

f:=aarctan13&plus;aarctan14&plus;barctan15

f:=aarctan13&plus;aarctan14&plus;barctan15

(9)

combinef&comma;arctan

aarctan711&plus;barctan15

(10)

combinearctanx&plus;arctan1xassumingx::real

12signumx&pi;

(11)

f:=arctanx&plus;arctany&colon;

combinef

arctanx&plus;arctany

(12)

combinef&comma;arctan&comma;symbolic

arctanx&plus;yxy&plus;1

(13)

assume0<x&comma;0<y

f&equals;combinef&comma;arctan

arctanx~&plus;arctany~&equals;arctanx~&plus;y~x~y~&plus;1&plus;&pi;

(14)

assumex<0&comma;0<y

f&equals;combinef&comma;arctan

arctanx~&plus;arctany~&equals;arctanx~&plus;y~x~y~&plus;1

(15)

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