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Elliptic Integral of the 1st Kind

- By Dr. Minas E. Lemonis, PhD - Updated: October 30, 2019

This tool evaluates the complete or incomplete elliptic integral of the first kind: K(k) or F(φ,k) respectively. Select the desired type of the calculation and enter the appropriate arguments below.

Type
k =
Result:

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About Elliptic Integrals of the First Kind

Definitions

The incomplete elliptic integral of the first kind is defined as:

\textrm{F}(\varphi,k) = \int_{0}^{\varphi}\frac{d\theta}{\sqrt{1-k^2\sin^2\theta}}

where k is the elliptic modulus, with -1 \le k \le 1 . Variable \varphi is the Jacobi's amplitude.

The complete elliptic integral of the first kind is defined as:

\textrm{K}(k) = \textrm{F}(\frac{\pi}{2},k) = \int_{0}^{\frac{\pi}{2}}\frac{d\theta}{\sqrt{1-k^2\sin^2\theta}}

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Values

In the following table the values of the complete elliptic integral of the first kind are shown for a range of k values:

kK(k)
-1
-0.9994.496
-0.993.357
-0.983.021
-0.972.828
-0.962.693
-0.952.59
-0.92.281
-0.81.995
-0.71.846
-0.61.751
-0.51.686
-0.41.64
-0.31.608
-0.21.587
-0.11.575
01.571
0.11.575
0.21.587
0.31.608
0.41.64
0.51.686
0.61.751
0.71.846
0.81.995
0.92.281
0.952.59
0.962.693
0.972.828
0.983.021
0.993.357
0.9994.496
1

In the following table the values of the incomplete elliptic integral of the first kind are shown for a range of k and φ values:

kF(30°,k)F(45°,k)F(60°,k)F(90°,k)F(135°,k)F(225°,k)F(270°,k)F(315°,k)
-10.54930.88141.3170.8814-0.8814-∞-0.8814
-0.9990.54920.88111.3164.4960.8811-0.8811-4.496-0.8811
-0.990.54870.87871.3073.3570.8787-0.8787-3.357-0.8787
-0.980.54820.87621.2973.0210.8762-0.8762-3.021-0.8762
-0.970.54760.87371.2872.8280.8737-0.8737-2.828-0.8737
-0.960.5470.87131.2792.6930.8713-0.8713-2.693-0.8713
-0.950.54650.86891.272.590.8689-0.8689-2.59-0.8689
-0.90.54390.85791.2332.2810.8579-0.8579-2.281-0.8579
-0.80.53930.83961.1791.9950.8396-0.8396-1.995-0.8396
-0.70.53540.82511.141.8460.8251-0.8251-1.846-0.8251
-0.60.53210.81351.1111.7510.8135-0.8135-1.751-0.8135
-0.50.52940.80441.091.6860.8044-0.8044-1.686-0.8044
-0.40.52730.79731.0731.640.7973-0.7973-1.64-0.7973
-0.30.52570.7921.0611.6080.792-0.792-1.608-0.792
-0.20.52450.78831.0531.5870.7883-0.7883-1.587-0.7883
-0.10.52380.78611.0491.5750.7861-0.7861-1.575-0.7861
00.52360.78541.0471.5710.7854-0.7854-1.571-0.7854
0.10.52380.78611.0491.5750.7861-0.7861-1.575-0.7861
0.20.52450.78831.0531.5870.7883-0.7883-1.587-0.7883
0.30.52570.7921.0611.6080.792-0.792-1.608-0.792
0.40.52730.79731.0731.640.7973-0.7973-1.64-0.7973
0.50.52940.80441.091.6860.8044-0.8044-1.686-0.8044
0.60.53210.81351.1111.7510.8135-0.8135-1.751-0.8135
0.70.53540.82511.141.8460.8251-0.8251-1.846-0.8251
0.80.53930.83961.1791.9950.8396-0.8396-1.995-0.8396
0.90.54390.85791.2332.2810.8579-0.8579-2.281-0.8579
0.950.54650.86891.272.590.8689-0.8689-2.59-0.8689
0.960.5470.87131.2792.6930.8713-0.8713-2.693-0.8713
0.970.54760.87371.2872.8280.8737-0.8737-2.828-0.8737
0.980.54820.87621.2973.0210.8762-0.8762-3.021-0.8762
0.990.54870.87871.3073.3570.8787-0.8787-3.357-0.8787
0.9990.54920.88111.3164.4960.8811-0.8811-4.496-0.8811
10.54930.88141.3170.8814-0.8814-∞-0.8814

The xy plots of the incomplete integral of the first kind for various values of amplitude φ are depicted in the following figure:

elliptic integral of the first kind xy plot/graph

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See also
Evaluate Elliptic Integral of the second kind
Evaluate Elliptic Integral of the third kind
Evaluate Carlson's form of Elliptic Integrals
All evaluation tools