Hille E.-an Integral Equality and Its Applications(1921)
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MA THEMA TICS: E . HILLE really greater f o r thick than f o r thin gold. I f t h e free path i s o f t h e order o f 10-lcm., that i s , comparable with t h e thickness o f t h e leaf, then t h e average path i n t h e thicker metal will b e longer than i n t h e thinner, a n d a greater departure would b e expected i n t h e thicker metal, a s found. [ I a m much indebted t o m assistant, M r. J. C . Slater, f o r h i s skill i n making t h e readings.] 1 Maxwell, C . , Everett, J . D . a n d Schuster, A . , B . A . Rep. 1876 (36-63). 2 Bridgman, P . W . , Physic. Rev., Ithaca, (2), 1 7 , 1921 (161-194). A N INTEGRAL EQUALITY A ND IT S APPLICATIONS B Y EINAR HILux HARVARD UNIvmRSiTY Communicated b y G . D . Birkhoff, M a y 1 5 , 1921 T h e purpose o f this note! i s to deduce a n integral equality adjoined t o a linear homogeneous differential equation o f t h e second order an d to show some applications o f such equalities t o the question o f t h e dis- tribution o f zeros o f solutions o f such differential equations i n t h e complex plane. L e t G(z) a n d K(z) b e t w o single-valued and analytic functions o f z throughout t he region under consideration a nd takethe differential equation ( 1 ) d-z[K)i(2dz ]+ G(z)w = O, o r t h e equivalent system d w W 2 d ( 2 ) {idt K(z)' with dW2= _G(z)wl, w2 = K(z)dW. From this system w e easily deduce t h e relation S I S 2 ( 3 ) [ww2]-w ( z) + 1w12G(z)dz = 0 , Si S i S l i where u denotes t h e conjugate o f u . If we p u t ( 4 ) dK = K d r = G(x)dz, =KG()dz relation ( 3 ) becomes 2 ; 2 ~~~~~2 i S ( 5 ) - 1W2] IW212 dK + fIwiI2dr = 0 , S i Si o r, split u p into real a n d imaginary parts, 3 0 3 OL,. 7, 1921
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