NUMERICAL EVALUATION OF A FOUR TERM ARCTAN FORMULA
ARCTAN FORMULA We have discussed earlier how one requires large values of N to obtain rapidly converging arctan (1/N) formulas for π One such formula which does this is the four term expression- π=48arctan(1/38) +80arctan(1/57) +28arctan(1/239) +96arctan(1/268) Let us demonstrate how one goes about evaluating this equality numerically
Arctangent Formulas and Pi - Grinnell College
Mathematical Assoc of America American Mathematical Monthly 121:1 August 4, 2018 2:23p m arctan˙2 tex page 2 where r is the radius of the inscribed circle Alternatively, applying Heron’s formula
PROPERTIES OF ARCTAN - University of Florida
This result relates the arctan to the logarithm function so that- 2 4 1 ln π i i = + Looking at the near linear relation between arctan(z) and z for z
Efficient Approximations for the Arctangent Function T
IEEE SIGNAL PROCESSING MAGAZINE [109] MAY 2006 arctan(x) ≈ π 4 x, −1 ≤ x ≤ 1 (2) This linear approximation has been used in [6] for FM demodulation due to its minimal complexity
The complex inverse trigonometric and hyperbolic functions
arccot(z) = arctan 1 z , (1) Arccot(z) = Arctan 1 z (2) Note that eqs (1) and (2) can be used as definitions of the inverse cotangent function and its principal value We now examine the principal value of the arccotangent for real-valued arguments Setting z = x, where x is real, Arccotx = 1 2 Arg x +i x − i 2
Integration Formulas - mathportalorg
www mathportal 2 2 2 2 2 2 2 2 2 2 2 arctan 4 0 4 4 1 2 2 4 ln 4 0 4 2 4 2 4 0 2 ax b for ac b ac b ac b ax b b ac dx for ac b ax bx c b ac ax b b ac for ac b
Derivative of arctan(x) - MIT OpenCourseWare
function of x, not of y We must now plug in the original formula for y, which was y = tan−1 x, to get y = cos2(arctan(x)) This is a correct answer but it can be simplified tremendously We’ll use some geometry to simplify it 1 x (1+x2)1/2 y Figure 3: Triangle with angles and lengths corresponding to those in the exam ple
Trigonometric Formula Sheet De nition of the Trig Functions
Trigonometric Formula Sheet De nition of the Trig Functions Right Triangle De nition Assume that: 0 <
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PROPERTIES OF ARCTAN(Z)
We know from elementary calculus that the function z=tan(ș) has an inverse ș=arctan(z). In differentiating z once we have- z zdzzequivalentitsorddz 0221)arctan(])tan(1[
On setting the upper limit to 1/N with N<1 we find the infinite series expansion for arctan given by- 01212/1
0 01 )12(1)1()1()/1arctan( nnnnN z nnNndzzN or the equivalent- dttt NnN N tm m n nn /1 0 221 0 12
1)12()1()/1arctan(
This series will converge quite rapidly when N>>1. Thus- ....)239(71 )239(51 )239(3112391)239/1arctan(642 However for N=1, the series just equals that of Gregory which is known to be notoriously slowly convergent-4.....91
7151
311)1arctan(
If one takes the first hundred terms(m=100) in the Gregory series, the integral remainder will still be- percentsomeordttt t3/1...0024999.01
1 0 2200In general the larger N becomes the more rapidly the infinite series for arctan(z) will converge. Thus the series for (ʌ/8) =arctan{ 1/[1+sqrt(2)]} reads - ...)21(1 )21(11)21(1 8 42
which converges somewhat faster than the Gregory series. Lets examine some of the other analytical characteristics of arctan(z). Its plot for z real looks like this- We see that arctan(z) varies linearly with z for small z starting with value zero and becomes non-linear in its variation with increasing z, eventually approaching Pi/2 as Pi/2-1/z as z approaches infinity. The function has odd symmetry since arctan(- z)=-arctan(z). Its derivative is just 1/(1+z^2) and hence represents a special case of the Witch of Agnesi ( this curve was studied by the Italian mathematician Maria Agnesi 1718-1799 and received its name due to a mistranslation of the Italian word versiero for curve by an English translator who mixed it up with the Italian word for witch). Using the multiple angle formula for tangent , one also has- or the equivalent - )arctan()arctan(])1()(arctan[yxxyyx
On setting x=z and y= we find -
)1arctan()arctan(2zz S so that, for example, arctan(2)=ʌ/2-arctan(0.5)=ʌ/2-0.46364..= 1.1071...If x=1 and y=-1/3 one obtains the well known identity- )31arctan()21arctan(4 S and x=1/7, y=-1/8 produces- )571arctan()81arctan()71arctan( Consider next the complex number z=x+iy. Writing this out in polar form yields- )]arctan(exp[ 22xyiyxiyx so that- 22
ln)/arctan( yxiyxixy This result relates the arctan to the logarithm function so that- 421ln
ii Looking at the near linear relation between arctan(z) and z for z<<1 suggests that arctan(1/N)=m*arctan(1/(m*N) +small correction of the order 1/N^3 for large N. This is indeed the case. By looking at the imaginary part of- )ln()ln()()(ln
221121
21iNpiNpiNiN pp one finds- ))34(1arctan()21arctan(2)1arctan( 2 NNNN )118278arctan()31arctan(3)1arctan( 24
NNN NN and- 242
NNNN NN We next solve an integral in terms of arctan to get-
Therefore one finds-
)117arctan(72)]73arctan()75[arctan(72 431 0 2 t ttdt It is also possible to manipulate the original integral form for artctan(z) into a variety of different range integrals. Consider the substitutions t=u/N and
Nt=tanh(v). These produce the integrals-
2222
22
42
arctan 42
44()2(1
bacbat bac abac abtdt a cbtatdt f 022210 22/1
0 2 )]11()[cosh(111)1arctan( vuN t N vdv NN uNduNdttN Expanding the term in the denominator of the last integral leads to an alternate series for arctan(1/N). In compact form, it reads- 022
2 )1()!12(!4
1)1arctan(nnnNnn
NN N and produces the identity- ...!92!4 !72!3 !52!2 !32!11242322212
Also using the variable substitution u=w/sqrt(w^2+1) yields the symmetric form- f f w w N wwdw NNN ]11)1[()1()1(2)/1arctan( 2222so that- 0022
21)(cosh()21)(cosh(2
1)5.0(
vv vdv vdv wwdw It seems that this last integral in w can form the starting point for an AGM approach for finding precise values of ʌ. It can also be expanded as the series- ...97531!47531!3
531!231!1
1!0 2 S which shows an interesting pattern but is unfortunately only slowly convergent. A much more rapidly convergent series is found for larger N. Indeed, we have in general that-quotesdbs_dbs7.pdfusesText_5