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1: 4.2 Definitions
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►This is a multivalued function of with branch point at .
►The principal value, or principal branch, is defined by
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►Most texts extend the definition of the principal value to include the branch cut
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►In all other cases, is a multivalued function with branch point at .
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►This result is also valid when has its principal value, provided that the branch of satisfies
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2: 6.15 Sums
3: 4.10 Integrals
4: 10.2 Definitions
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§10.2(ii) Standard Solutions
… ►The principal branch corresponds to the principal branches of in (10.2.3) and (10.2.4), with a cut in the -plane along the interval . … ►Each solution has a branch point at for all . … ► … ►Branch Conventions
…5: 28.7 Analytic Continuation of Eigenvalues
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►The only singularities are algebraic branch points, with and finite at these points.
The number of branch points is infinite, but countable, and there are no finite limit points.
…The branch points are called the exceptional values, and the other points normal values.
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►For a visualization of the first branch point of and see Figure 28.7.1.
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6: 4.6 Power Series
7: 4.13 Lambert -Function
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►We call the increasing solution for which the principal branch and denote it by .
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►Other solutions of (4.13.1) are other branches of .
… is a single-valued analytic function on , real-valued when , and has a square root branch point at .
…The other branches
are single-valued analytic functions on , have a logarithmic branch point at , and, in the case , have a square root branch point at respectively.
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►where for , for on the relevant branch cuts,
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8: 4.37 Inverse Hyperbolic Functions
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►Elsewhere on the integration paths in (4.37.1) and (4.37.2) the branches are determined by continuity.
… and have branch points at ; the other four functions have branch points at .
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►The principal values (or principal branches) of the inverse , , and are obtained by introducing cuts in the -plane as indicated in Figure 4.37.1(i)-(iii), and requiring the integration paths in (4.37.1)–(4.37.3) not to cross these cuts.
…The principal branches are denoted by , , respectively.
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9: 4.3 Graphics
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