These formulas can even serve as the definition of the trigonometric functions for complex arguments x.

These results, however, are very sensitive to the accuracy of the twiddle factors used in the FFT ( i. e. the trigonometric function values ), and it is not unusual for incautious FFT implementations to have much worse accuracy, e. g. if they use inaccurate trigonometric recurrence formulas.

These simple formulas will not do if the input values are not equally likely, or are not independent.

These two researchers developed several formulas for mineral nutrient solutions, known as Hoagland solution.

These same interpolation formulas nevertheless continue to be used as part of the software algorithms for solving differential equations.

These formulas provide the solution for the initial-value problem for the wave equation.

These changes can be represented in formulas by inserting appropriate electrons into each half-reaction:

These formulas however only provide the calculation for the τ = τ < sub > 0 </ sub > case.

These formulas may be used when h is much smaller than the radius of the Earth ( 6371 km ), including all views from any mountaintops, aeroplanes, or high-altitude balloons.

These formulas characterize the periapsis and apoapsis of an orbit:

These formulas are traditionally used in analog televisions and equipment ; digital equipment such as HDTV and digital video cameras use Y ' CbCr.

These formulas are connected to the Schubert decomposition of the Grassmannian, and are q-analogs of the Betti numbers of complex Grassmannians.

These formulas are the basis for comparing and investigating the nature of non-random patterns in the structure of food web networks among many different types of ecosystems.

These formulas are generalized by the Faà di Bruno's formula.

These formulas can also be written using Lagrange's notation.

These formulas contain the " slope " of the line connecting P and Q, hence involve divisions between residue classes modulo n, which can be performed using the extended Euclidean algorithm.

These operators are direct extensions of their complex analogs: if and are taken from the real subset of complex numbers, the appearance of the conjugate in the formulas has no effect, so the operators are the same as those for the complex numbers.

These ideas were regarded as dynamic forces that Herbart attempted to explain by means of mathematical formulas.

These formulas for denatured alcohol are found in 27 CFR part 21 of the Code of Federal Regulations.

These formulas are for single-frequency sine waves or narrowband signals.

These formulas only state in another form the assertions in the table of derivatives.

These formulas all provide a different view into the processes that generate longshore drift.

These five formulas are:

These two formulas, added together, produce

These vehicles, with large sensitive areas, have collected data for long enough times to give reliable impact rates for the periods of exposure.

These values give direction to the life of the middle-class man or woman, dictating the methods of child rearing, determining the pattern of community participation, setting the style for the psychiatric treatment of middle-class illness, and informing the congregational life of the major denominations.

These give a characteristic " laddered " appearance on agar gel after electrophoresis.

These coatings are long gone and much of the architectural detail is sadly eroded, though detached fragments found in the ruins during consolidation give an impression of the original refined, rather austere, architectural effect.

These political divisions subdivide the population of Ajaccio into units that can be more democratically served but they do not give a true picture of the size of Ajaccio.

These included, among others, the "... saints who live on rays which moon and daystar give " and " those ... whose food the wave of air supplies ".

These two bodies, to which we can give or from which we can remove the heat without causing their temperatures to vary, exercise the functions of two unlimited reservoirs of caloric.

These give them an advantage in combat since the limbs are outfitted with offensive weapons.

These alternative mating strategies give females more mating opportunities without losing the support of the males in their community.

These will enable developing countries to give their children a digital education, and help close the digital divide across the world.

These arguments are expressed in a variety of different but equivalent ways ( they give the same elliptic integral ).

These restrictions effectively give the British Government the ability to prevent the island's government from declaring the islands to be a tax haven or from establishing a central bank.

These give more specific instructions on how to apply statutes, which often explicitly delegate regulation of specific details to the government.

These combined factors of toed-in rail fins cause several issues: drag on a toed-in outside rail fin can slow the board down in trim, but it can also give a braking effect during turns that is useful.

These four schools share most of their rulings, but differ on the particular hadiths they accept as authentic and the weight they give to analogy or reason ( qiyas ) in deciding difficulties.

These systems give law enforcement personnel real-time, street-level intelligence that allows them to not only focuses on the worst graffiti offenders and their damage, but also to monitor potential gang violence that is associated with the graffiti.

These studies can also give useful information about pathways for metamorphism through pressure, temperature, space, and time.

These were carefully cut to what is approximately a face slope with a seked of 5½ palms to give the required dimensions.

These properties, which give glass its clearness, can be retained even if glass is partially light-absorbing i. e. colored.

* Linear algebraic groups ( or more generally affine group schemes ) — These are the analogues of Lie groups, but over more general fields than just R or C. Although linear algebraic groups have a classification that is very similar to that of Lie groups, and give rise to the same families of Lie algebras, their representations are rather different ( and much less well understood ).

These numbers are highly speculative ; many sources give vague ideas of the unemployment rating being ( for example, in 2003 ) around 50 %, giving the impression that the actual rate could be several percentage points higher or lower.

These people ( or their descendants ) could exchange their vouchers for land previously owned by agricultural co-operatives, who were forced to give up some of their land for this purpose.

These notions ( packing dimension, Hausdorff dimension, Minkowski – Bouligand dimension ) all give the same value for many shapes, but there are well documented exceptions.

These letters give a glimpse into the daily trials of ruling an empire, from dealing with floods and mandating changes to a flawed calendar, to taking care of Babylon's massive herds of livestock.

These anions can be partially deprotonated to give the bicarbonate ( HCO < sub > 3 </ sub >< sup >−</ sup >) and hydrogenoxalate ( HC < sub > 2 </ sub > O < sub > 4 </ sub >< sup >−</ sup >).