- #1
electron2
- 46
- 0
[tex]\frac{1}{s(s^2-1)}=\frac{A}{s}+\frac{b}{s-1}+\frac{C}{s+1}[/tex]
i get A=-1
B=-C
A+B+C=0
where is the mistake
i get A=-1
B=-C
A+B+C=0
where is the mistake
electron2 said:[tex]\frac{1}{s(s^2-1)}=\frac{A}{s}+\frac{b}{s-1}+\frac{C}{s+1}[/tex]
i get A=-1
B=-C
A+B+C=0
where is the mistake
Count Iblis said:You can do it faster, without solving any equations, by expanding around the poles. A rational function f(z) with a pole at z = p can be expanded as:
a/(z-p)^n + b/(z-p)^(n-1) + ...
If you add up all the singular terms of all the expansions around all the poles, and call that function you get g(z), then the difference:
h(z) = f(z) - g(z)
doesn't have any poles anymore, therefore it must be a polynomial. If
f(z) tends to zero at inifinty, then so does h(z), because g(z) also tends to zero to infinity. But because h(z) is a polynomial, that means that h(z) = 0, therefore f(z) = g(z).
In the given case, the expansions around the three poles is very easy to obtain. Around s = 0, we have:
f(s) = 1/s [expansion of 1/(s^2 - 1) around s = 0] =
1/s [-1 + O(s)] = -1/s + nonsingular terms.
Around s = 1:
f(s) = 1/(s-1) [expansion of 1/s 1/(s+1) around s = 1] =
1/2 1/(s-1) + nonsingular terms
Around s = -1:
f(s) = 1/(s+1) [expansion of 1/s 1/(s-1) around s = -1] =
1/2 1/(s+1) + nonsingular terms
So, we have:
f(s) = -1/s + 1/2 [1/(s-1) + 1/(s+1)]
The best method for breaking a material into simple fractures depends on several factors, such as the type of material, its thickness, and its level of brittleness. Some common methods include using a hammer and chisel, applying pressure with a hydraulic press, or using specialized tools such as a diamond saw. It is important to research and carefully consider the properties of the material before selecting a method.
Safety should always be a top priority when breaking a material into simple fractures. It is important to wear appropriate protective gear, such as safety glasses and gloves, and to work in a well-ventilated area. It is also important to carefully follow the instructions for the chosen method and to have a first aid kit on hand in case of accidents.
The angle and direction of the fracture will depend on the type of material and the method being used. In general, it is best to start at the edges of the material and work towards the center, applying controlled force to create a clean break. Experiment with different angles and directions to find the most effective approach for the specific material.
Chemical agents can be used to weaken or dissolve certain types of materials, making them easier to break into simple fractures. However, it is important to use caution and carefully follow the instructions for handling and disposing of these chemicals. Additionally, the use of chemical agents may not be suitable for all materials and may require specialized knowledge and equipment.
If the material does not break into simple fractures as desired, it is important to stop and assess the situation. It may be necessary to adjust the method being used or to seek assistance from a professional. It is also important to carefully examine the material for any potential hazards, such as sharp edges or flying debris, before continuing the process.