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A common choice of branch cut is the negative real axis, although the choice is largely a matter of convenience. The logarithm has a jump discontinuity of 2 π i when crossing the branch cut. The logarithm can be made continuous by gluing together countably many copies, called sheets, of the complex plane along the branch cut. On each sheet ...
Any number log z defined by such criteria has the property that e log z = z. In this manner log function is a multi-valued function (often referred to as a "multifunction" in the context of complex analysis). A branch cut, usually along the negative real axis, can limit the imaginary part so it lies between −π and π. These are the chosen ...
Industrial applications of cutting-stock problems for high production volumes arise especially when basic material is produced in large rolls that are further cut into smaller units (see roll slitting). This is done e.g. in paper and plastic film industries but also in production of flat metals like steel or brass.
A single branch of the complex logarithm. The hue of the color is used to show the argument of the complex logarithm. The brightness of the color is used to show the modulus of the complex logarithm. The real part of log(z) is the natural logarithm of | z |. Its graph is thus obtained by rotating the graph of ln(x) around the z-axis.
However, the important thing to note is that z 1/2 = e (Log z)/2, so z 1/2 has a branch cut. This affects our choice of the contour C. Normally the logarithm branch cut is defined as the negative real axis, however, this makes the calculation of the integral slightly more complicated, so we define it to be the positive real axis.
There are four types of materials available: O-1 tool steel, A-2 tool steel, A-6 tool steel, and 1018 steel (low-carbon or low-carb steel). Lengths are either 18 or 36 in (457 or 914 mm) long, various widths up to 16 in (406 mm) are available, and thicknesses range from 1 ⁄ 64 to 2.875 in (0.40 to 73.03 mm).
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Chisel — a sharpened blade for cutting roots, digging and prying. A pinch point bar has a chisel at one end. Bars are typically 5 to 6 ft (1.5 to 1.8 m) long and weigh 15 to 23 lb (6.8 to 10.4 kg). They are usually made entirely of cylindrical or hexagonal forged steel with a diameter of approximately 1 in (2.5 cm).