In this paper a perturbation method is developed for treating non‐standard propagation of microwaves beyond the horizon in the case when the deviation of theM‐curve from the standard (≡ theM‐anomaly) can be represented by a term&agr;e−&lgr;z, wherezdenotes height in natural units. HereMdenotes the modified index of refraction of the air. The method is also applicable to other forms of theM‐anomaly which can be derived from an exponential term by differentiation with respect to &lgr;; in fact, in its region of convergence it is formally applicable to the most general type ofM‐curve, including elevated ducts. The region of practical convergence of the method ranges from highly substandard conditions down to cases where the decrement is a fraction of the standard value. The procedure followed is to express the height‐gain functionUk(z) of thekth mode in the non‐standard case as a linear combination of the height‐gain functionsUm0(z) of all the modes in the standard case:Uk(z)=m=1∞AkmUm0(z).The execution of this plan hinges on the possibility of evaluating the quantities&bgr;nm(&lgr;)=0∞Un0(z)Um0(z)e−&lgr;zdz.It is shown that &bgr;nm(&lgr;) satisfies the differential equationd&bgr;nmd&lgr;=22&lgr;+&bgr;nm(&lgr;).−12&lgr;+12(Dm0+Dn0)+&lgr;24+14&lgr;2(Dm0−Dn0)2,whose solution is&bgr;nm(&lgr;)=12√&lgr;exp&lgr;2(Dn0+Dm0)+&lgr;312−14&lgr;(Dm0−Dn0)2·0&lgr;dx√xexp−x2(Dm0+Dn0)−x312+14x(Dn0−Dm0)2·HereDm0denotes the characteristic value of themth mode in the standard case. For large &lgr; the following asymptotic formula holds&bgr;nm=−2&lgr;3+2&lgr;(Dm0+Dn0)−2+1&lgr;(Dm0−Dn0)2+83&lgr;3+2&lgr;(Dm0+Dn0)−1&lgr;(Dm0−Dn0)2&lgr;3+2&lgr;(Dm0+Dn0)−2+1&lgr;(Dm0−Dn0)23·Having determined the &bgr;nm(&lgr;) from (d), or by a numerical solution of (c), the characteristic valuesDkand the coefficientsAkmare to be solved from the infinite system of equationsm=1∞Akm[(Dk−Dm0)&dgr;nm+&agr;&bgr;nm(&lgr;)]=0, n=1,2,3,…,where &dgr;nmdenotes the Kronecker symbol. For this purpose a simple iterative procedure has been developed, which has been found to be rapidly convergent. TheAkmare normalized by the condition0∞Uk2(z)dz=1=m=1∞Akm2.[The integral ∫0∞Uk2(z)dzdiverges when taken along the real axis; it converges, however, and to the same limit, when the path is a radial line in the fourth quadrant of thezplane. In the sequel, whenever an integral is divergent, it will be understood that the path is suitably modified.] One can also expandDkas a power series in &agr;Dk=Dk(0)+&agr;Dk(1)+&agr;2Dk(2)+…,Dk(1)=−&bgr;kk; Dk(2)=ei&pgr;/3 m&bgr;mk2(&tgr;m−&tgr;k),m≠k.An alternative expression forDk(2)is given in Eq. (58).