Geodesic
On one equality of integrals
Dalʹnevostočnyj matematičeskij žurnal, Tome 22 (2022) no. 1, pp. 51-54.

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The note proves the equality $$\frac1{2\pi i}\mskip-8mu\oint_{|z|=r}\mskip-8mu\ln\mskip-4mu\left[1{-}f(z)\left(z{+}\frac1z\right)\right]dz= -\frac1{2\pi i}\mskip-8mu\oint_{|z|=r}\mskip-4mu\exp\left(\frac{1{-}\sqrt{1-4f^2(z)}}{2zf(z)}\right)dz.$$ 
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A. A. Dmitriev. On one equality of integrals. Dalʹnevostočnyj matematičeskij žurnal, Tome 22 (2022) no. 1, pp. 51-54. http://geodesic.mathdoc.fr/item/DVMG_2022_22_1_a3/

[1] by А. П. Прудников, Ю. А. Брычков, О. И. Маричев Integraly i ryady, v. 1, Elementarnye funktsii, FIZMATLIT, M., 2002, 632 pp. | MR

[2] G. N. Vatson, Teoriya besselevykh funktsii, IIL, M., 1949, 799 pp.