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If the Laurent series has only finitely many terms with negative powers of z - c, then the singularity is a pole. The biggest negative exponent is the order of the pole.
Example: Singularities of rational functions with no common factors in its numerator and denominator. (These come from setting the denominator equal to 0.)
If the Laurent series has infinitely many terms with negative powers of z - c, then the singularity is essential.
Example: e^(1/z) = 1 + (1/z) + (1/2!) 1/z^2 + ... has an essential singularity at z = 0.
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What is the title of this picture in Punchline Algebra book b 13.15?
The photo is called "Fire pole used for false alarms."
What is the sage mode?
Sage mode is a mode where the user collects and uses Natural Energy. Normally, chakra is made of 2 elements, Spiritual (mental) energy, and Physical energy, and its quality is determined by how well the two are balanced.During Sage mode, Natural Energy is inserted as well. When mastered, Sage mode will not only make your techniques around 100 times as strong, it will also protect your body from injury (Naruto survived without a scratch a drop from about 200m high pole), and would extend your hit range (You don't have to actually land a hit, if you miss a little, it would still inflict damage as if it was a direct hit.).A fully empowered sage, who is also capable of Merging with a sage frog, will never get tired during battle, because Natural Energy never runs out.Definitely a scary ability, but with all great power come great risks. Too little Natural Energy, and you won't be able to use Sage Mode, too much, and you turn into a frog (wtf?!).
