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If the principal part of the Laurent’s series vanishes, then the Laurent’s series reduces to
- a)Cauchy’s series
- b)Maclaurin’s series
- c)Taylor’s series
- d)None of the above
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
If the principal part of the Laurent’s series vanishes, then the...
Taylor Series:
If f(z) is analytic inside a circle 'C', centre at z = a, and radius 'r', then for all z inside 'C'; the Taylor series is given by-
Laurent Series:
If f(z) is analytic at every point inside and on the boundary of a ring shaped region 'R' bounded by two concentric circle C1 and C2 having centre at 'a' & respective radii r1 and r2 (r1 > r2).
The negative part of Laurent's series i.e
is called the singular part, and if that vanishes the terms that remain will be 
, which is nothing but Taylor series.
Laurent Series:
If f(z) is analytic at every point inside and on the boundary of a ring shaped region 'R' bounded by two concentric circle C1 and C2 having centre at 'a' & respective radii r1 and r2 (r1 > r2).
The negative part of Laurent's series i.e
is called the singular part, and if that vanishes the terms that remain will be , which is nothing but Taylor series.Most Upvoted Answer
If the principal part of the Laurent’s series vanishes, then the...
Taylor Series:
If f(z) is analytic inside a circle 'C', centre at z = a, and radius 'r', then for all z inside 'C'; the Taylor series is given by-
Laurent Series:
If f(z) is analytic at every point inside and on the boundary of a ring shaped region 'R' bounded by two concentric circle C1 and C2 having centre at 'a' & respective radii r1 and r2 (r1 > r2).
The negative part of Laurent's series i.eis called the singular part, and if that vanishes the terms that remain will be , which is nothing but Taylor series.
Laurent Series:
If f(z) is analytic at every point inside and on the boundary of a ring shaped region 'R' bounded by two concentric circle C1 and C2 having centre at 'a' & respective radii r1 and r2 (r1 > r2).
The negative part of Laurent's series i.e
is called the singular part, and if that vanishes the terms that remain will be , which is nothing but Taylor series.Free Test
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Community Answer
If the principal part of the Laurent’s series vanishes, then the...
Laurent's Series Reduction
When the principal part of the Laurent's series vanishes, it means that the series contains only the analytic part, which is the Taylor series. Therefore, the Laurent's series reduces to Taylor's series in such cases.
Explanation
- Laurent's Series: Laurent's series is a representation of a complex function as a series of two parts - an analytic part (Taylor series) and a principal part (containing negative powers of the variable).
- Principal Part Vanishes: When the principal part of the Laurent's series vanishes, it implies that there are no terms with negative powers of the variable in the series.
- Taylor's Series: Taylor's series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point (usually the center of the series).
- Reduction: Since the principal part disappears from the Laurent's series, the series simplifies to contain only the Taylor series part, leading to a reduction to Taylor's series.
Therefore, if the principal part of the Laurent's series vanishes, the Laurent's series reduces to Taylor's series. This reduction occurs due to the absence of terms with negative powers of the variable, leaving only the analytic part of the series.
When the principal part of the Laurent's series vanishes, it means that the series contains only the analytic part, which is the Taylor series. Therefore, the Laurent's series reduces to Taylor's series in such cases.
Explanation
- Laurent's Series: Laurent's series is a representation of a complex function as a series of two parts - an analytic part (Taylor series) and a principal part (containing negative powers of the variable).
- Principal Part Vanishes: When the principal part of the Laurent's series vanishes, it implies that there are no terms with negative powers of the variable in the series.
- Taylor's Series: Taylor's series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point (usually the center of the series).
- Reduction: Since the principal part disappears from the Laurent's series, the series simplifies to contain only the Taylor series part, leading to a reduction to Taylor's series.
Therefore, if the principal part of the Laurent's series vanishes, the Laurent's series reduces to Taylor's series. This reduction occurs due to the absence of terms with negative powers of the variable, leaving only the analytic part of the series.
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If the principal part of the Laurent’s series vanishes, then the Laurent’s series reduces toa)Cauchy’s seriesb)Maclaurin’s seriesc)Taylor’s seriesd)None of the aboveCorrect answer is option 'C'. Can you explain this answer?
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