Approximations of elliptic integrals

Elliptic integrals are not expressed in terms of elementary functions. By definition, elementary functions ^[1] are functions defined by formulas containing a finite number of algebraic or trigonometric operations performed on an argument, function, and some constants.

Elliptic integrals in the Legendre form of the 1st, 2nd, and 3rd kind ^[1] , as well as integrals similar to them (with the substitution of plus signs for minus signs and / or with the replacement of cos with sin or vice versa) are precisely represented by the functional series . Such a representation is not an elementary function due to the infinite number of members of this series.

Guided by considerations of achieving the necessary accuracy and taking into account the n initial members of the series and neglecting the remainder, that is, the sum of the remaining members of the series from n + 1 to $\infty$ ${\ displaystyle \ infty}$ $\ infty$ , we obtain an approximation of the (definite or indefinite) elliptic integral in the form of an elementary function. Approximations of elliptic integrals are applied similarly to ordinary integrals .

A certain integral of the first kind can be represented as:

\int _{\varphi _{1}}^{\varphi _{2}}{\frac {d\varphi }{\sqrt {1-k^{2}\sin ^{2}\varphi }}}={\sqrt {1+E}}\left(\varphi -{\frac {E\sin 2\varphi }{4}}+...\right){\Bigr |}_{\varphi _{1}}^{\varphi _{2}};

{\ displaystyle \ int _ {\ varphi _ {1}} ^ {\ varphi _ {2}} {\ frac {d \ varphi} {\ sqrt {1-k ^ {2} \ sin ^ {2} \ varphi }}} = {\ sqrt {1 + E}} \ left (\ varphi - {\ frac {E \ sin 2 \ varphi} {4}} + ... \ right) {\ Bigr |} _ {\ varphi _ {1}} ^ {\ varphi _ {2}};}

{\ displaystyle \ int _ {\ varphi _ {1}} ^ {\ varphi _ {2}} {\ frac {d \ varphi} {\ sqrt {1-k ^ {2} \ sin ^ {2} \ varphi }}} = {\ sqrt {1 + E}} \ left (\ varphi - {\ frac {E \ sin 2 \ varphi} {4}} + ... \ right) {\ Bigr |} _ {\ varphi _ {1}} ^ {\ varphi _ {2}};}

$(\varepsilon \approx 4{,}2\cdot 10^{-6};~~\mu _{\varepsilon }(2)\approx 330).$ ${\ displaystyle (\ varepsilon \ approx 4 {,} 2 \ cdot 10 ^ {- 6}; ~~ \ mu _ {\ varepsilon} (2) \ approx 330).}$ ${\ displaystyle (\ varepsilon \ approx 4 {,} 2 \ cdot 10 ^ {- 6}; ~~ \ mu _ {\ varepsilon} (2) \ approx 330).}$

Hereinafter in the formulas the following notation applies:

E={\frac {k^{2}}{2-k^{2}}};~~~N={\frac {h}{2+h}};

{\ displaystyle E = {\ frac {k ^ {2}} {2-k ^ {2}}}; ~~~ N = {\ frac {h} {2 + h}};}

{\ displaystyle E = {\ frac {k ^ {2}} {2-k ^ {2}}}; ~~~ N = {\ frac {h} {2 + h}};}

\varepsilon _{0}=\left|{\frac {F(\varphi )\mid _{\varphi _{1}}^{\varphi _{2}}-\int _{\varphi _{1}}^{\varphi _{2}}f(\varphi )d\varphi }{\int _{\varphi _{1}}^{\varphi _{2}}f(\varphi )d\varphi }}\right|

{\ displaystyle \ varepsilon _ {0} = \ left | {\ frac {F (\ varphi) \ mid _ {\ varphi _ {1}} ^ {\ varphi _ {2}} - \ int _ {\ varphi _ {1}} ^ {\ varphi _ {2}} f (\ varphi) d \ varphi} {\ int _ {\ varphi _ {1}} ^ {\ varphi _ {2}} f (\ varphi) d \ varphi}} \ right |}

{\ displaystyle \ varepsilon _ {0} = \ left | {\ frac {F (\ varphi) \ mid _ {\ varphi _ {1}} ^ {\ varphi _ {2}} - \ int _ {\ varphi _ {1}} ^ {\ varphi _ {2}} f (\ varphi) d \ varphi} {\ int _ {\ varphi _ {1}} ^ {\ varphi _ {2}} f (\ varphi) d \ varphi}} \ right |}

Is the calculated relative error in the calculation of elliptic integrals using the indicated formulas for ellipses similar to the Earth’s meridional (

k^{2}=0{,}006693

{\ displaystyle k ^ {2} = 0 {,} 006693}

{\ displaystyle k ^ {2} = 0 {,} 006693}

and

h=0{,}006674

{\ displaystyle h = 0 {,} 006674}

{\ displaystyle h = 0 {,} 006674}

)

\varepsilon =\varepsilon _{0max}

{\ displaystyle \ varepsilon = \ varepsilon _ {0max}}

{\ displaystyle \ varepsilon = \ varepsilon _ {0max}}

- maximum calculated relative error of the corresponding formula in the range of angles

\Delta ~\varphi =\varphi _{2}-\varphi _{1}<{\frac {\pi }{2}}.

{\ displaystyle \ Delta ~ \ varphi = \ varphi _ {2} - \ varphi _ {1} <{\ frac {\ pi} {2}}.}

{\ displaystyle \ Delta ~ \ varphi = \ varphi _ {2} - \ varphi _ {1} <{\ frac {\ pi} {2}}.}

\mu _{\varepsilon }(m)

{\ displaystyle \ mu _ {\ varepsilon} (m)}

{\ displaystyle \ mu _ {\ varepsilon} (m)}

- a number indicating how many times the maximum calculated relative error of the corresponding formula will decrease, if we add

m

{\ displaystyle m}

m

unspecified members in her decomposition formula.

A definite integral of the second kind can be represented as:

\int _{\varphi _{1}}^{\varphi _{2}}{\sqrt {1-k^{2}\sin ^{2}\varphi }}\ d\varphi ={\frac {1}{\sqrt {1+E}}}\left(\varphi +{\frac {E\sin 2\varphi }{4}}+...\right){\Bigr |}_{\varphi _{1}}^{\varphi _{2}};

{\ displaystyle \ int _ {\ varphi _ {1}} ^ {\ varphi _ {2}} {\ sqrt {1-k ^ {2} \ sin ^ {2} \ varphi}} \ d \ varphi = { \ frac {1} {\ sqrt {1 + E}}} \ left (\ varphi + {\ frac {E \ sin 2 \ varphi} {4}} + ... \ right) {\ Bigr |} _ { \ varphi _ {1}} ^ {\ varphi _ {2}};}

{\ displaystyle \ int _ {\ varphi _ {1}} ^ {\ varphi _ {2}} {\ sqrt {1-k ^ {2} \ sin ^ {2} \ varphi}} \ d \ varphi = { \ frac {1} {\ sqrt {1 + E}}} \ left (\ varphi + {\ frac {E \ sin 2 \ varphi} {4}} + ... \ right) {\ Bigr |} _ { \ varphi _ {1}} ^ {\ varphi _ {2}};}

$(\varepsilon \approx 1{,}4\cdot 10^{-6};~~\mu _{\varepsilon }(2)\approx 500).$ ${\ displaystyle (\ varepsilon \ approx 1 {,} 4 \ cdot 10 ^ {- 6}; ~~ \ mu _ {\ varepsilon} (2) \ approx 500).}$ ${\ displaystyle (\ varepsilon \ approx 1 {,} 4 \ cdot 10 ^ {- 6}; ~~ \ mu _ {\ varepsilon} (2) \ approx 500).}$

The length of the ellipse arc with a single semi-major axis:

\int _{\varphi _{1}}^{\varphi _{2}}{\sqrt {1-k^{2}\cos ^{2}\varphi }}\ d\varphi ={\frac {1}{\sqrt {1+E}}}\left(\varphi -{\frac {E\sin 2\varphi }{4}}+...\right){\Bigr |}_{\varphi _{1}}^{\varphi _{2}};

{\ displaystyle \ int _ {\ varphi _ {1}} ^ {\ varphi _ {2}} {\ sqrt {1-k ^ {2} \ cos ^ {2} \ varphi}} \ d \ varphi = { \ frac {1} {\ sqrt {1 + E}}} \ left (\ varphi - {\ frac {E \ sin 2 \ varphi} {4}} + ... \ right) {\ Bigr |} _ { \ varphi _ {1}} ^ {\ varphi _ {2}};}

{\ displaystyle \ int _ {\ varphi _ {1}} ^ {\ varphi _ {2}} {\ sqrt {1-k ^ {2} \ cos ^ {2} \ varphi}} \ d \ varphi = { \ frac {1} {\ sqrt {1 + E}}} \ left (\ varphi - {\ frac {E \ sin 2 \ varphi} {4}} + ... \ right) {\ Bigr |} _ { \ varphi _ {1}} ^ {\ varphi _ {2}};}

$(\varepsilon \approx 1{,}4\cdot 10^{-6};~~\mu _{\varepsilon }(2)\approx 500).$ ${\ displaystyle (\ varepsilon \ approx 1 {,} 4 \ cdot 10 ^ {- 6}; ~~ \ mu _ {\ varepsilon} (2) \ approx 500).}$ ${\ displaystyle (\ varepsilon \ approx 1 {,} 4 \ cdot 10 ^ {- 6}; ~~ \ mu _ {\ varepsilon} (2) \ approx 500).}$

A certain integral of the 3rd kind can be written in the form:

\int _{\varphi _{1}}^{\varphi _{2}}{\frac {d\varphi }{(1+h\cdot \sin ^{2}\varphi ){\sqrt {1-k^{2}\sin ^{2}\varphi }}}}=

{\ displaystyle \ int _ {\ varphi _ {1}} ^ {\ varphi _ {2}} {\ frac {d \ varphi} {(1 + h \ cdot \ sin ^ {2} \ varphi) {\ sqrt {1-k ^ {2} \ sin ^ {2} \ varphi}}}} =}

{\ displaystyle \ int _ {\ varphi _ {1}} ^ {\ varphi _ {2}} {\ frac {d \ varphi} {(1 + h \ cdot \ sin ^ {2} \ varphi) {\ sqrt {1-k ^ {2} \ sin ^ {2} \ varphi}}}} =}

={\frac {\sqrt {1+E}}{2+h}}\cdot \left({\frac {2+h}{\sqrt {1+h}}}\operatorname {arctg} \left({\sqrt {1+h}}\cdot \operatorname {tg} \varphi \right)\left(1-{\frac {E}{2N}}+...\right)+\varphi \cdot \left({\frac {E}{N}}+...\right)+...\right){\Bigl |}_{\varphi _{1}}^{\varphi _{2}};

{\ displaystyle = {\ frac {\ sqrt {1 + E}} {2 + h}} \ cdot \ left ({\ frac {2 + h} {\ sqrt {1 + h}}} \ operatorname {arctg} \ left ({\ sqrt {1 + h}} \ cdot \ operatorname {tg} \ varphi \ right) \ left (1 - {\ frac {E} {2N}} + ... \ right) + \ varphi \ cdot \ left ({\ frac {E} {N}} + ... \ right) + ... \ right) {\ Bigl |} _ {\ varphi _ {1}} ^ {\ varphi _ {2} };}

{\ displaystyle = {\ frac {\ sqrt {1 + E}} {2 + h}} \ cdot \ left ({\ frac {2 + h} {\ sqrt {1 + h}}} \ operatorname {arctg} \ left ({\ sqrt {1 + h}} \ cdot \ operatorname {tg} \ varphi \ right) \ left (1 - {\ frac {E} {2N}} + ... \ right) + \ varphi \ cdot \ left ({\ frac {E} {N}} + ... \ right) + ... \ right) {\ Bigl |} _ {\ varphi _ {1}} ^ {\ varphi _ {2} };}

(\varepsilon \approx 4{,}2\cdot 10^{-6};~~\mu _{\varepsilon }(3)\approx 330).

{\ displaystyle (\ varepsilon \ approx 4 {,} 2 \ cdot 10 ^ {- 6}; ~~ \ mu _ {\ varepsilon} (3) \ approx 330).}

{\ displaystyle (\ varepsilon \ approx 4 {,} 2 \ cdot 10 ^ {- 6}; ~~ \ mu _ {\ varepsilon} (3) \ approx 330).}

Example

To calculate the length of the arc of the geodesic line on the surface of the terrestrial spheroid ^[2] , the calculation of a certain integral of the form is required:

\int _{\varphi _{1}}^{\varphi _{2}}{\frac {d\varphi }{(1+h\cdot \cos ^{2}\varphi ){\sqrt {1-k^{2}\sin ^{2}\varphi }}}}=

{\ displaystyle \ int _ {\ varphi _ {1}} ^ {\ varphi _ {2}} {\ frac {d \ varphi} {(1 + h \ cdot \ cos ^ {2} \ varphi) {\ sqrt {1-k ^ {2} \ sin ^ {2} \ varphi}}}} =}

={\frac {\sqrt {1+E}}{2+h}}\cdot \left({\frac {2+h}{\sqrt {1+h}}}{\text{arctg}}\left({\frac {{\text{tg}}\ \varphi }{\sqrt {1+h}}}\right)\left(1+{\frac {E}{2N}}+...\right)-\varphi \left({\frac {E}{N}}+...\right)+...\right){\Bigr |}_{\varphi _{1}}^{\varphi _{2}};

{\ displaystyle = {\ frac {\ sqrt {1 + E}} {2 + h}} \ cdot \ left ({\ frac {2 + h} {\ sqrt {1 + h}}} {\ text {arctg }} \ left ({\ frac {{\ text {tg}} \ \ varphi} {\ sqrt {1 + h}}} \ right) \ left (1 + {\ frac {E} {2N}} +. .. \ right) - \ varphi \ left ({\ frac {E} {N}} + ... \ right) + ... \ right) {\ Bigr |} _ {\ varphi _ {1}} ^ {\ varphi _ {2}};}

(\varepsilon \approx 4{,}2\cdot 10^{-6};~~\mu _{\varepsilon }(3)\approx 330).

{\ displaystyle (\ varepsilon \ approx 4 {,} 2 \ cdot 10 ^ {- 6}; ~~ \ mu _ {\ varepsilon} (3) \ approx 330).}

Notes

↑ ¹ ² Bronstein I.N., Semendyaev K.A. Handbook of Mathematics. - M.: Science, 1964.
↑ Matsevich M.I. Navigation calculations of geodesic routes. - M.: Information Fund FSUE "VNTIC", No. 72200700019, 2007.

[sprav-1] ¹ ² Bronstein I.N., Semendyaev K.A. Handbook of Mathematics. - M.: Science, 1964.

[geo-2] Matsevich M.I. Navigation calculations of geodesic routes. - M.: Information Fund FSUE "VNTIC", No. 72200700019, 2007.

Approximations of elliptic integrals

Example

See also

Notes

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