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<TITLE>CLHS: Section 12.1.5.4</TITLE>
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12.1.5.4 Principal Values and Branch Cuts</H2> <P>
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Many of the irrational and transcendental functions are multiply defined in the complex domain; for example, there are in general an infinite number of complex values for the logarithm function. In each such case, a <A REL=DEFINITION HREF="26_glo_p.htm#principal"><I>principal</I></A> <A REL=DEFINITION HREF="26_glo_v.htm#value"><I>value</I></A> must be chosen for the function to return. In general, such values cannot be chosen so as to make the range continuous; lines in the domain called branch cuts must be defined, which in turn define the discontinuities in the range. Common Lisp defines the branch cuts, <A REL=DEFINITION HREF="26_glo_p.htm#principal"><I>principal</I></A> <A REL=DEFINITION HREF="26_glo_v.htm#value"><I>values</I></A>, and boundary conditions for the complex functions following ``Principal Values and Branch Cuts in Complex APL.'' The branch cut rules that apply to each function are located with the description of that function. <P>
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The next figure lists the identities that are obeyed throughout the applicable portion of the complex domain, even on the branch cuts: <P>
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<PRE>
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sin i z = i sinh z sinh i z = i sin z arctan i z = i arctanh z
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cos i z = cosh z cosh i z = cos z arcsinh i z = i arcsin z
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tan i z = i tanh z arcsin i z = i arcsinh z arctanh i z = i arctan z
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</PRE>
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<P><B>Figure 12-9. Trigonometric Identities for Complex Domain</B> <P>
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The quadrant numbers referred to in the discussions of branch cuts are as illustrated in the next figure. <P>
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<P><IMG ALIGN=CENTER ALT="[Quadrants image]" WIDTH=335 HEIGHT=165 SRC="../Graphics/Quadrant.gif"><P> <P><B>Figure 12-10. Quadrant Numbering for Branch Cuts</B> <P>
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