Кіріспе
Функцияның айналасында қалыпсыз мінез-құлық байқалатын орын. Нақты мәнді функциялардың маңызды ерекшеліктері.
Location around which a function displays irregular behavior
essential singularities of real valued functions
essential singularities of real valued functions
Күрделі талдауда функцияның маңызды ерекшелігі – функцияның өте күшті мінез-құлық көрсететін "қиын" ерекшелігі. Маңызды сингулярлық санаты – басқа тәсілдермен шешілетін ерекшеліктерге жатпайтын, басқаруға өте қиын оқшауланған сингулярлықтардың "қалдық" тобы. Іс жүзінде кейбіреулер оқшауланбаған ерекшеліктерді де қамтиды, онда қалдық болмайды.
In complex analysis, an essential singularity of a function is a "severe" singularity near which the function exhibits striking behavior. The category essential singularity is a "left over" or default group of isolated singularities that are especially unmanageable: by definition they fit into neither of the other two categories of singularity that may be dealt with in some manner – removable singularities and poles. In practice some include non isolated singularities too; those do not have a residue.
Ресми сипаттама
Комплекс жазықтығының ашық ішкі жиынын және голоморфтық функцияның элементі деп қарастырайық. Егер нүктесіндегі сингулярлық полюс немесе алынып тасталатын сингулярлық болмаса, онда бұл нүкте функцияның негізгі сингулярлығы деп аталады. Мысалы, функциясы нүктесінде негізгі сингулярлыққа ие.
Consider an open subset of the complex plane Let be an element of , and a holomorphic function. The point is called an essential singularity of the function if the singularity is neither a pole nor a removable singularity. For example, the function has an essential singularity at .
Балама сипаттамалар
Болсын – кешенді сан, және болсын функциясы нүктеде анықталмаған, бірақ кешенді жазықтықтың кейбір аймағында аналитикалық, және әрбір ашық маңда нүктемен қиылысады.
Let be a complex number, and assume that is not defined at but is analytic in some region of the complex plane, and that every open neighbourhood of has non empty intersection with
If both and exist, then is a removable singularity of both and
If exists but does not exist (in fact ), then is a zero of and a pole of
Similarly, if does not exist (in fact ) but exists, then is a pole of and a zero of
If neither nor exists, then is an essential singularity of both and
Another way to characterize an essential singularity is that the Laurent series of at the point has infinitely many negative degree terms (i. e., the principal part of the Laurent series is an infinite sum). A related definition is that if there is a point for which no derivative of converges to a limit as tends to , then is an essential singularity of
On a Riemann sphere with a point at infinity, , the function has an essential singularity at that point if and only if the has an essential singularity at 0: i. e. neither nor exists. The Riemann zeta function on the Riemann sphere has only one essential singularity, at Indeed, every meromorphic function aside that is not a rational function has a unique essential singularity at
The behavior of holomorphic functions near their essential singularities is described by the Casorati–Weierstrass theorem and by the considerably stronger Picard's great theorem. The latter says that in every neighborhood of an essential singularity , the function takes on every complex value, except possibly one, infinitely many times. (The exception is necessary; for example, the function never takes on the value 0.)
If both and exist, then is a removable singularity of both and
If exists but does not exist (in fact ), then is a zero of and a pole of
Similarly, if does not exist (in fact ) but exists, then is a pole of and a zero of
If neither nor exists, then is an essential singularity of both and
Another way to characterize an essential singularity is that the Laurent series of at the point has infinitely many negative degree terms (i. e., the principal part of the Laurent series is an infinite sum). A related definition is that if there is a point for which no derivative of converges to a limit as tends to , then is an essential singularity of
On a Riemann sphere with a point at infinity, , the function has an essential singularity at that point if and only if the has an essential singularity at 0: i. e. neither nor exists. The Riemann zeta function on the Riemann sphere has only one essential singularity, at Indeed, every meromorphic function aside that is not a rational function has a unique essential singularity at
The behavior of holomorphic functions near their essential singularities is described by the Casorati–Weierstrass theorem and by the considerably stronger Picard's great theorem. The latter says that in every neighborhood of an essential singularity , the function takes on every complex value, except possibly one, infinitely many times. (The exception is necessary; for example, the function never takes on the value 0.)
Егер екі жақ та бар болса, онда функциялардың екеуі үшін де алынып тасталатын сингулярлық болады.
Let be a complex number, and assume that is not defined at but is analytic in some region of the complex plane, and that every open neighbourhood of has non empty intersection with
If both and exist, then is a removable singularity of both and
If exists but does not exist (in fact ), then is a zero of and a pole of
Similarly, if does not exist (in fact ) but exists, then is a pole of and a zero of
If neither nor exists, then is an essential singularity of both and
Another way to characterize an essential singularity is that the Laurent series of at the point has infinitely many negative degree terms (i. e., the principal part of the Laurent series is an infinite sum). A related definition is that if there is a point for which no derivative of converges to a limit as tends to , then is an essential singularity of
On a Riemann sphere with a point at infinity, , the function has an essential singularity at that point if and only if the has an essential singularity at 0: i. e. neither nor exists. The Riemann zeta function on the Riemann sphere has only one essential singularity, at Indeed, every meromorphic function aside that is not a rational function has a unique essential singularity at
The behavior of holomorphic functions near their essential singularities is described by the Casorati–Weierstrass theorem and by the considerably stronger Picard's great theorem. The latter says that in every neighborhood of an essential singularity , the function takes on every complex value, except possibly one, infinitely many times. (The exception is necessary; for example, the function never takes on the value 0.)
If both and exist, then is a removable singularity of both and
If exists but does not exist (in fact ), then is a zero of and a pole of
Similarly, if does not exist (in fact ) but exists, then is a pole of and a zero of
If neither nor exists, then is an essential singularity of both and
Another way to characterize an essential singularity is that the Laurent series of at the point has infinitely many negative degree terms (i. e., the principal part of the Laurent series is an infinite sum). A related definition is that if there is a point for which no derivative of converges to a limit as tends to , then is an essential singularity of
On a Riemann sphere with a point at infinity, , the function has an essential singularity at that point if and only if the has an essential singularity at 0: i. e. neither nor exists. The Riemann zeta function on the Riemann sphere has only one essential singularity, at Indeed, every meromorphic function aside that is not a rational function has a unique essential singularity at
The behavior of holomorphic functions near their essential singularities is described by the Casorati–Weierstrass theorem and by the considerably stronger Picard's great theorem. The latter says that in every neighborhood of an essential singularity , the function takes on every complex value, except possibly one, infinitely many times. (The exception is necessary; for example, the function never takes on the value 0.)
Егер бар болса, бірақ жоқ болса (әрине), онда функцияның нүлі және полюсі болады. Сол сияқты, егер жоқ болса (әрине), бірақ бар болса, онда функцияның полюсі және нүлі болады.
Let be a complex number, and assume that is not defined at but is analytic in some region of the complex plane, and that every open neighbourhood of has non empty intersection with
If both and exist, then is a removable singularity of both and
If exists but does not exist (in fact ), then is a zero of and a pole of
Similarly, if does not exist (in fact ) but exists, then is a pole of and a zero of
If neither nor exists, then is an essential singularity of both and
Another way to characterize an essential singularity is that the Laurent series of at the point has infinitely many negative degree terms (i. e., the principal part of the Laurent series is an infinite sum). A related definition is that if there is a point for which no derivative of converges to a limit as tends to , then is an essential singularity of
On a Riemann sphere with a point at infinity, , the function has an essential singularity at that point if and only if the has an essential singularity at 0: i. e. neither nor exists. The Riemann zeta function on the Riemann sphere has only one essential singularity, at Indeed, every meromorphic function aside that is not a rational function has a unique essential singularity at
The behavior of holomorphic functions near their essential singularities is described by the Casorati–Weierstrass theorem and by the considerably stronger Picard's great theorem. The latter says that in every neighborhood of an essential singularity , the function takes on every complex value, except possibly one, infinitely many times. (The exception is necessary; for example, the function never takes on the value 0.)
If both and exist, then is a removable singularity of both and
If exists but does not exist (in fact ), then is a zero of and a pole of
Similarly, if does not exist (in fact ) but exists, then is a pole of and a zero of
If neither nor exists, then is an essential singularity of both and
Another way to characterize an essential singularity is that the Laurent series of at the point has infinitely many negative degree terms (i. e., the principal part of the Laurent series is an infinite sum). A related definition is that if there is a point for which no derivative of converges to a limit as tends to , then is an essential singularity of
On a Riemann sphere with a point at infinity, , the function has an essential singularity at that point if and only if the has an essential singularity at 0: i. e. neither nor exists. The Riemann zeta function on the Riemann sphere has only one essential singularity, at Indeed, every meromorphic function aside that is not a rational function has a unique essential singularity at
The behavior of holomorphic functions near their essential singularities is described by the Casorati–Weierstrass theorem and by the considerably stronger Picard's great theorem. The latter says that in every neighborhood of an essential singularity , the function takes on every complex value, except possibly one, infinitely many times. (The exception is necessary; for example, the function never takes on the value 0.)
Егер екі жақ та болмаса, онда функциялардың екеуі үшін де маңызды сингулярлық болады. Маңызды сингулярлықты сипаттаудың тағы бір жолы – функцияның Лоран қатары нүктеде шексіз көп теріс дәрежелі мүшелерге ие (яғни, Лоран қатарының негізгі бөлігі шексіз жиынтық). Бұған байланысты анықтама: егер функцияның туындысы шекке жақындамайтын нүкте болса, онда функцияның нүктеде маңызды сингулярлығы болады.
Let be a complex number, and assume that is not defined at but is analytic in some region of the complex plane, and that every open neighbourhood of has non empty intersection with
If both and exist, then is a removable singularity of both and
If exists but does not exist (in fact ), then is a zero of and a pole of
Similarly, if does not exist (in fact ) but exists, then is a pole of and a zero of
If neither nor exists, then is an essential singularity of both and
Another way to characterize an essential singularity is that the Laurent series of at the point has infinitely many negative degree terms (i. e., the principal part of the Laurent series is an infinite sum). A related definition is that if there is a point for which no derivative of converges to a limit as tends to , then is an essential singularity of
On a Riemann sphere with a point at infinity, , the function has an essential singularity at that point if and only if the has an essential singularity at 0: i. e. neither nor exists. The Riemann zeta function on the Riemann sphere has only one essential singularity, at Indeed, every meromorphic function aside that is not a rational function has a unique essential singularity at
The behavior of holomorphic functions near their essential singularities is described by the Casorati–Weierstrass theorem and by the considerably stronger Picard's great theorem. The latter says that in every neighborhood of an essential singularity , the function takes on every complex value, except possibly one, infinitely many times. (The exception is necessary; for example, the function never takes on the value 0.)
If both and exist, then is a removable singularity of both and
If exists but does not exist (in fact ), then is a zero of and a pole of
Similarly, if does not exist (in fact ) but exists, then is a pole of and a zero of
If neither nor exists, then is an essential singularity of both and
Another way to characterize an essential singularity is that the Laurent series of at the point has infinitely many negative degree terms (i. e., the principal part of the Laurent series is an infinite sum). A related definition is that if there is a point for which no derivative of converges to a limit as tends to , then is an essential singularity of
On a Riemann sphere with a point at infinity, , the function has an essential singularity at that point if and only if the has an essential singularity at 0: i. e. neither nor exists. The Riemann zeta function on the Riemann sphere has only one essential singularity, at Indeed, every meromorphic function aside that is not a rational function has a unique essential singularity at
The behavior of holomorphic functions near their essential singularities is described by the Casorati–Weierstrass theorem and by the considerably stronger Picard's great theorem. The latter says that in every neighborhood of an essential singularity , the function takes on every complex value, except possibly one, infinitely many times. (The exception is necessary; for example, the function never takes on the value 0.)
Riemann сферасында шексіз нүктесі бар жағдайда, функция сол нүктеде маңызды сингулярлыққа ие, егер және тек егер 0 нүктесінде маңызды сингулярлық болса: яғни, екі жақ та болмаса. Riemann сферасындағы Riemann zeta функциясының тек бір ғана маңызды сингулярлығы бар, шындығында, рационалды функция емес әрбір мероморфтық функцияның бірегей маңызды сингулярлығы бар.
Let be a complex number, and assume that is not defined at but is analytic in some region of the complex plane, and that every open neighbourhood of has non empty intersection with
If both and exist, then is a removable singularity of both and
If exists but does not exist (in fact ), then is a zero of and a pole of
Similarly, if does not exist (in fact ) but exists, then is a pole of and a zero of
If neither nor exists, then is an essential singularity of both and
Another way to characterize an essential singularity is that the Laurent series of at the point has infinitely many negative degree terms (i. e., the principal part of the Laurent series is an infinite sum). A related definition is that if there is a point for which no derivative of converges to a limit as tends to , then is an essential singularity of
On a Riemann sphere with a point at infinity, , the function has an essential singularity at that point if and only if the has an essential singularity at 0: i. e. neither nor exists. The Riemann zeta function on the Riemann sphere has only one essential singularity, at Indeed, every meromorphic function aside that is not a rational function has a unique essential singularity at
The behavior of holomorphic functions near their essential singularities is described by the Casorati–Weierstrass theorem and by the considerably stronger Picard's great theorem. The latter says that in every neighborhood of an essential singularity , the function takes on every complex value, except possibly one, infinitely many times. (The exception is necessary; for example, the function never takes on the value 0.)
If both and exist, then is a removable singularity of both and
If exists but does not exist (in fact ), then is a zero of and a pole of
Similarly, if does not exist (in fact ) but exists, then is a pole of and a zero of
If neither nor exists, then is an essential singularity of both and
Another way to characterize an essential singularity is that the Laurent series of at the point has infinitely many negative degree terms (i. e., the principal part of the Laurent series is an infinite sum). A related definition is that if there is a point for which no derivative of converges to a limit as tends to , then is an essential singularity of
On a Riemann sphere with a point at infinity, , the function has an essential singularity at that point if and only if the has an essential singularity at 0: i. e. neither nor exists. The Riemann zeta function on the Riemann sphere has only one essential singularity, at Indeed, every meromorphic function aside that is not a rational function has a unique essential singularity at
The behavior of holomorphic functions near their essential singularities is described by the Casorati–Weierstrass theorem and by the considerably stronger Picard's great theorem. The latter says that in every neighborhood of an essential singularity , the function takes on every complex value, except possibly one, infinitely many times. (The exception is necessary; for example, the function never takes on the value 0.)
Голоморфтық функциялардың маңызды сингулярлықтарына жақын мінез-құлқы Казорати–Вейерштрасс теоремасымен және одан да күшті Пикардтың үлкен теоремасымен сипатталады. Соңғысы, функцияның әрбір маңында маңызды сингулярлық, мүмкін бірін қоспағанда, әрбір кешенді мәнді шексіз көп рет қабылдайды. (Ерекшелік қажет; мысалы, функция ешқашан 0 мәнін қабылдамайды.)
Let be a complex number, and assume that is not defined at but is analytic in some region of the complex plane, and that every open neighbourhood of has non empty intersection with
If both and exist, then is a removable singularity of both and
If exists but does not exist (in fact ), then is a zero of and a pole of
Similarly, if does not exist (in fact ) but exists, then is a pole of and a zero of
If neither nor exists, then is an essential singularity of both and
Another way to characterize an essential singularity is that the Laurent series of at the point has infinitely many negative degree terms (i. e., the principal part of the Laurent series is an infinite sum). A related definition is that if there is a point for which no derivative of converges to a limit as tends to , then is an essential singularity of
On a Riemann sphere with a point at infinity, , the function has an essential singularity at that point if and only if the has an essential singularity at 0: i. e. neither nor exists. The Riemann zeta function on the Riemann sphere has only one essential singularity, at Indeed, every meromorphic function aside that is not a rational function has a unique essential singularity at
The behavior of holomorphic functions near their essential singularities is described by the Casorati–Weierstrass theorem and by the considerably stronger Picard's great theorem. The latter says that in every neighborhood of an essential singularity , the function takes on every complex value, except possibly one, infinitely many times. (The exception is necessary; for example, the function never takes on the value 0.)
If both and exist, then is a removable singularity of both and
If exists but does not exist (in fact ), then is a zero of and a pole of
Similarly, if does not exist (in fact ) but exists, then is a pole of and a zero of
If neither nor exists, then is an essential singularity of both and
Another way to characterize an essential singularity is that the Laurent series of at the point has infinitely many negative degree terms (i. e., the principal part of the Laurent series is an infinite sum). A related definition is that if there is a point for which no derivative of converges to a limit as tends to , then is an essential singularity of
On a Riemann sphere with a point at infinity, , the function has an essential singularity at that point if and only if the has an essential singularity at 0: i. e. neither nor exists. The Riemann zeta function on the Riemann sphere has only one essential singularity, at Indeed, every meromorphic function aside that is not a rational function has a unique essential singularity at
The behavior of holomorphic functions near their essential singularities is described by the Casorati–Weierstrass theorem and by the considerably stronger Picard's great theorem. The latter says that in every neighborhood of an essential singularity , the function takes on every complex value, except possibly one, infinitely many times. (The exception is necessary; for example, the function never takes on the value 0.)