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- 2 8(r)

(sphere)

(4.1.38)

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where PS stands for principal value with an infinitesimal volume of defined shape excluded about the point l' = O. The expression (sphere) at the end of (4.1.38) indicates that a spherical infinitesimal volume has been chosen for the shape of the exclusion. The coefficient in front of the Dirac delta function of the second term depends on the shape of the exclusion volume. If shapes other than spherical are chosen, then the coefficient will be different from that of (4.1.38). For a spherical exclusion we have

=(0)

(4.1.39)

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where (4.1.40) (4.1.41) Using (4.1.39) through (4.1.41) and the spherically symmetric correlation functions,

Using the exponential correlation function as in the scalar case, we obtain the following in the low-frequency limit of kma 1 J 2 . 2 K = k n - 3k n + 3Jk~a2(1 + 2'lk m a) (4.1.43) The result in (4.1.43) is valid if the zeroth-order solution [the first term in (4.1.43)] is much larger than the first-order solution. Thus the validity conditions of (4.1.43) are J l and (4.1.44) (4.1.45)

=(0)

Arrange, ArrangeOverride, CaptureMouse, CheckAccess, Equals, Finalize, FindName, GetAnimationBaseValue, GetHashCode, GetType, GetValue, HitTest, InvalidateArrange, InvalidateMeasure, Measure, MeasureOverride, MemberwiseClone, Pause, Play, ReleaseMouseCapture, SetBinding, SetSource, SetValue, Stop, ToString, TransformToVisual, UpdateLayout

(4.1.42)

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The maximum order or accuracy of a method is either k + 1 or k + 2 depending on whether k is odd or even, respectively Partial stability occurs when the step length (h) is critical to the solution and is particularly relevant when considering Runge-Kutta methods In general, the roots zi of (IV27) are dependent on the product hi, and also on equations (IV26) The stability boundary is the value of hi for which lzil = 1, and any method which has this boundary is termed conditionally stable A method with an infinite stability boundary is known as A-stable (unconditionally stable) A linear multistep method is A-stable if all solutions of (IV25) tend to zero as n + co when the method is applied with fixed h > 0 to (IV23) where j is a complex constant with Re(;.

Note that (4.1.45) is identical to (4.1.35) of the scalar case. However, the condition (4.1.44) is new and is not in the scalar wave case. Thus the vector wave case has one more condition than the scalar wave case. Equation (4.1.44) assumes that the variance of permittivity fluctuations is small. The weak permittivity fluctuation criterion of (4.1.44) is restrictive in applications because many media are characterized by large permittivity fluctuations. Physically, in cases of large permittivity fluctuations, the internal field inside an inhomogeneity can be quite different from that of the background

medium. The restriction in (4.1.44) can be eliminated by taking into account the singularity of the dyadic Green's function. The incorporation of the singularity will be done in Section 3 on strong permittivity fluctuations.

BufferingProgressChanged, CurrentStateChanged, DownloadProgressChanged, GotFocus, KeyDown, KeyUp, LayoutUpdated, Loaded, LostFocus, MarkerReached, MediaEnded, MediaFailed, MediaOpened, MouseEnter, MouseLeave, MouseLeftButtonDown, MouseLeftButtonUp, MouseMove, SizeChanged

The terms in the mass operator can be rearranged to form a new renormalized mass operator. Consider the diagram: ~

(4.1.46)

Equation (4.1.46) reproduces all strongly connected diagrams in the Neumann series for the mean field which have one outer correlation and both strongly and weakly connected diagrams inside. Next, observe that all the strongly connected diagrams with two-crossed outer correlations may be summed by the following

ActualHeight, ActualWidth, AspectRatio, Clip, Cursor, DataContext, DesiredSize, Dispatcher, Height, HorizontalAlignment, IsHitTestVisible, Language, Margin, MaxHeight, MaxWidth, MinHeight, MinWidth, Name, Opacity, OpacityMask, Parent, RenderSize, RenderTransform, RenderTransformOrigin, Resources, Source, Style, SubImages, Tag, Triggers, UseSprings, VerticalAlignment, ViewportOrigin, ViewportWidth, Visibility, Width

(4.1.47)

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