4. Implicit function
1. Single variable implicit function
For by the equation
F ( x , y )=0
The determined implicit function has the following theorem :
[ Existence Theorem ] Let the function F ( x , y ) be defined in a certain neighborhood * R of the point M _{0} ( x _{0} , y _{0} ) and satisfy the following conditions :
(i) F ( x , y ) and its partial derivatives are continuous in R ,_{}
(ii) F ( x _{0} , y _{0} )=0,
(iii) ≠ 0,_{}
Then in some neighborhood of point M _{0} ( x _{0} , y _{0} )
_{}_{}; )_{}
There is a unique singlevalued function y = f ( x ) in it , which has the following properties :
1° F [ x , f ( x )] ≡ 0, and f ( x _{0} ) = y _{0} ,
2° The function f ( x ) is continuous in the interval ( ) ,_{}
3° It has continuous derivatives in this interval ._{}
[ Calculation of Derivatives ]
_{} ( ≠ 0)_{}
_{} ( ≠ 0)_{}
2. Multivariate implicit function
For by the equation
F ( x , y , z )=0
The determined implicit function has the following theorem :
[ Existence Theorem ] Let the function F ( x , y , z ) be defined in a certain neighborhood R of the point P _{0} ( x _{0} , y _{0} , z _{0} ) and satisfy the following conditions :
(i) F ( x , y , z ) and its partial derivatives , continuous in R ,_{}_{}
(ii) F ( x _{0} , y _{0} , z _{0} )=0, _{}_{}_{}
(iii) ( x _{0} , y _{0} , z _{0} ) ≠ 0,_{}_{}_{}_{}
Then in some neighborhood of point P _{0} ( x _{0} , y _{0} , z _{0} )
_{}_{}; ; )_{}_{}
There exists a unique singlevalued function z = h ( x , y ) with the following properties :
1° F [ x , y , h ( x , y )] ≡ 0, and h ( x _{0} , y _{0} ) = z _{0} ,
The 2° function h ( x , y ) is continuous ,
3° It has continuous partial derivatives ._{}
[ Calculation of Derivatives ]
_{}, ( ≠ 0)_{} _{}
If you need to find all the first , second , _{}and partial derivatives of each order , just replace the identity
F ( x , y , z )=0
Find the first  order , second ^{} order , third  order ... _ _ _ _ _^{}_{}
Note that for the equation
F ( x _{1} , , x _{n} , y )=0_{}_{}
The identified implicit functions have similar results .
3. Implicit function determined by the system of equations
pair by the system of equations
_{} (1)
The determined implicit function has the following theorem :
[ Existence Theorem ] Let the functions F ( x , y , z ) and G ( x , y , z ) be defined in a certain neighborhood R of the point P 0 ( _{x} 0 , _{y} 0 , _{z} 0 ) _{and} satisfy the following conditions :
(i) F ( x , y , z ), G ( x , y , z ) and all their partial derivatives are continuous in R ,
(ii) F ( x _{0} , y _{0} , z _{0} )=0, G ( x _{0} , y _{0} , z _{0} )=0, _{}_{}_{}_{}_{}_{}
(iii) Determinant
J ( x , y , z )=_{}
At point P _{0} ( x _{0} , y _{0} , z _{0} ) is not equal to zero : J ( x _{0} , y _{0} , z _{0} ) ≠ 0.
Then in some neighborhood of point P _{0} ( x _{0} , y _{0} , z _{0} )
_{}_{}; ; )_{}_{}
There is a unique set of singlevalued functions y = f ( x ), z = g ( x ) in , with the following properties :
1° F [ x , f ( x ), g ( x )] ≡ 0, G [ x , f ( x ), g ( x )] ≡ 0, and f ( x _{0} ) = y _{0} , g ( x _{0} )= z _{0} ,
2° In the interval ( ) , the functions f ( x ), g ( x ) are continuous ,_{}
3° has continuous derivatives in this interval ._{}
[ Calculation of Derivatives ] Consider y and z as implicit functions of x , and differentiate equation system (1) with respect to x to get
_{}
This is a system of linear equations with respect to and whose determinant J ≠ 0, from which and can be solved ._{}_{}_{}_{}
Note that for the system of equations
_{}
The identified implicit functions have similar results .
5. Variable Substitution in Differential Expressions
1. Univariate function
Let y = f ( x ), and have an expression containing the independent variable, the dependent variable, and their derivatives
H = F ( x , y , )_{}
When used as variable substitution , each derivative can be calculated as follows:
[ In the case of independent variable transformation ] Let the transformation formula be
x =_{}
At this time , _{}_{}
_{} (1)
………………
[ The case where both the independent variable and the function are transformed ] Let the transformation formula be
x = , y =_{}_{}
where t is the new independent variable and u is the new function .
At this time , by the differential law of the composite function, we get
_{},_{}
_{}
_{}
……………………
Substitute these formulas into formula (1) to get the result .
2. Multivariate functions
[ In the case of independent variable transformation ] Let z = f ( x , y ), and there is an expression containing independent variables, dependent variables and their partial derivatives
H = F ( x , y , z , , , ,…)_{}_{}_{}
The transformation formula is
x = , y =_{}_{}
where u and sum are new independent variables , then the partial derivative is determined by the following equation :_{}_{}_{}
_{}= +_{}_{}_{}_{}
_{}
Other higherorder partial derivatives can also be calculated in this way .
[ The case where both the independent variable and the function are transformed ] Let the transformation formula be
x = , y = , z =_{}_{}_{}
where u , _{}is the new independent variable , w = w ( u , v ) is the new function , then the partial derivative is determined by the following equation :_{}_{}
_{}_{}+ )+ + )= +_{}_{}_{}_{}_{}_{}_{}_{}
_{}
Other higherorder partial derivatives can also be calculated in this way .
Note that when H is not an individual partial derivative , but all the partial derivatives of the given order , it is more convenient to use the total differential when calculating the successive partial derivatives .
6. The Fundamental Theorem of Differential Calculus ( Mean Value Theorem )

[ Lore's Theorem ] If (i) the function f ( x ) is defined on the closed interval [ a , b ] and is continuous , (ii) there is a finite derivative in the open interval ( a , b ) , and (iii) there is a finite derivative in the interval ( a , b ) The function values are equal at both ends of : f ( a ) = f ( b ). Then there is at least a point c between a and b , so that =0. That is, the curve y = f ( x ) is at point ( c , _{} _{} The tangent at f ( c )) is horizontal ( Fig. 5.6).
In particular , if f ( a ) = f ( b ) = 0, Lohr's theorem can be formulated as follows : between two roots of a function , its first derivative has at least one root .
Note that the function f ( x ) must be continuous on the closed interval [ a , b ] , and there must be a derivative in the open interval ( a , b ) , which is very important for the correctness of the conclusion of the theorem . For example, the function
f ( x ) = On the interval [0, 1] , all conditions of the theorem are satisfied except for the discontinuity when x = 1 , but = 1 everywhere in (0, 1) . For example, by the equation f ( The function defined by x ) = x ( ) and f ( x ) = ( ) also satisfies all the conditions of the theorem except when x = the ( bilateral ) derivative does not exist in this interval , but the derivative is in the left half It is equal to +1 in the interval and equal to +1 in the right half of the interval . _{}_{} _{} _{}_{}_{}_{}_{}
The condition (iii) of the theorem is also very important , for example, the function f ( x )= x on the interval [0,1] satisfies all the conditions of the theorem except condition (iii) , and its derivative is everywhere = 1. _{}
[ Mean Value Theorem ] If (i) f ( x ) is defined on the closed interval [ a , b ] and is continuous, and (ii) there is a finite derivative in the open interval ( a , b ) , then between a and b There is at least one point c between them , which satisfies the equation _{}
_{}= ( a < c < b ) (1)_{}
Figure 5.7 
That is, the tangent of the curve y = f ( x ) at the point ( c , f ( c )) is parallel to the chord AB ( Figure 5.7). This theorem is also known as the finite change theorem or Lagrange's theorem .
(1) is also often written in the following forms :
f ( b ) _{}
f ( x + Δ x ) Δ x ( x < c < x + Δ x ) _{}
Δ y = f ( x + Δ x ) ( ) _{} _{}
From the mean value theorem we get
Theorem If every point on the interval [ a , b ] has =0, then the function f ( x ) is a constant on this interval . _{}
[ Cauchy's theorem ] If (i) the functions f ( t ) and g ( t ) are continuous in the closed interval [ a , b ] , (ii) have finite derivatives in the open interval ( a , b ) , and (iii) are in the open interval ( a , b ) In the interval ( a , b ) ≠ 0. Then there is at least a point c between a and b such that_{}
Figure 5.8 
_{}= ( a < c < b )_{}
This formula is called Cauchy's formula ( Figure 5.8). Cauchy's theorem is often called the generalized mean value theorem of differential calculus , since when g ( t ) = x , this formula is formula (1).
[ Mean Value Theorem for Multivariable Functions ] If (i) the function f ( x , y ) is defined over a closed region and is continuous , (ii) there are continuous partial derivatives inside this region ( ie, at all its interior points ) , , now examine two points in D_{}_{}_{}
M _{0} ( x _{0} , y _{0} ) and M _{1} ( x _{0} + Δ x , y _{0} + Δ y )
Assuming that these two points can be connected by straight line segments M _{0 }M _{1 all located in the }D area , the following formula holds :_{}_{}
Δ f ( x _{0} , y _{0} ) = f ( x _{0} + Δ x , y _{0} + Δ y )_{}
= (0< θ < 1)_{}_{}
From the mean value theorem we get
Theorem If the continuous function f ( x , y ) in the closed connected region D* , the partial derivatives in this region are all equal to zero , that is
_{}= =0,_{}
Then this function must be constant in region D.
7. Taylor formula and Taylor series
1. Taylor's formula for univariate functions
[ Taylor's local formula ] If the function f ( x ) satisfies the conditions : (i) it is defined in a certain neighborhood of point a , (ii) has a derivative up to order in this neighborhood , , (iii) has at point a The n order derivative , then f ( x ) can be expressed in the following forms in the neighborhood of point a :_{}_{}_{}_{}_{}
1° f ( a + h )= f ( a )+ _{}
* If any two points in the area can be connected by a "polyline", and all the points of the polyline are in this area, this area is called a connected area .
= ( as h → 0)_{}
2° f ( x )= f ( a )+ _{}
= ( when x → a )_{}
In particular , when a = 0 , there are
[ Maclaurin formula ]
f ( x )= f (0)+ _{}
= ( as x → 0)_{}
[ Taylor formula ] If the function f ( x ) satisfies the conditions : (i) defined on the closed interval [ a , b ] , (ii) there is a continuous derivative up to the nth order on this closed interval , (iii) when a < There is a finite derivative when x < b , then f ( x ) can be expressed in the following forms on the closed interval [ a , b ] : _{}_{}_{}
1° f ( a + h )= ( a < a + h < b )_{}
where R _{n} ( h ) = (0 < θ < 1) ( Lagrange remainder ) _{}_{}
or R _{n} ( h ) = (0 < θ < 1) ( Cauchytype remainder ) _{}_{}
2° f ( x )= ( _{} _{})
where R _{n} ( x ) = ( a < ξ < b ) ( Lagrangian remainder ) _{}_{}
or R _{n} ( x ) = (0 < θ < 1) ( Cauchytype remainder ) _{}_{}
In particular , when a = 0 , there are
[ Maclaurin formula ]
f ( x )= ( _{} _{})
where R _{n} ( x ) = ( a < ξ < b ) ( Lagrangian remainder ) _{}_{}
or R _{n} ( x ) = (0 < θ < 1) ( Cauchytype remainder ) _{}_{}
[ Taylor series ] In Taylor's formula 2° with remainder , if the expansion is raised to an arbitrarily high power of ( ) _{}, we have
f ( x )= f ( a )+_{}
Whether it converges or not , and whether its sum equals f ( x ), is called the Taylor series of the function f ( x ) . The coefficients of the power of ( )_{}
f ( a ), , ,…, ,…_{}_{}_{}
called the Taylor coefficient .
[ Maclaurin series ] In the Maclaurin formula with remainder , if the expansion proceeds to an arbitrarily high power of x , we have
f ( x )= f (0)+_{}
Whether it converges or not , and whether its sum equals f ( x ), is called the Maclaurin series of the function f ( x ) . Coefficients of powers of x
f (0), , ,…, ,… _{}_{}_{}
called the Maclaurin coefficient .
For the Taylor formula of polynomials ( Qin Jiushao's method ), see Chapter 3 , §2, 1 .
2. Taylor's formula for multivariate functions
[ Taylor formula ] Assume that the binary function f ( x , y ) in the neighborhood D of a certain point ( x _{0} , y _{0} ) has all continuous partial derivatives up to the n +1 order . Give x and y a change h and k , so that the straight line segment connecting the points ( x _{0} , y _{0} ) and ( x _{0} + h , y _{0} + k ) does not go beyond D ,_{}_{}_{}_{}Then f ( x , y ) can be expressed in D in the form :
1° f ( x _{0} + h , y _{0} + k )= _{}_{}_{}
(0< θ < 1)
The symbol in the formula_{}
The meaning is as follows : treat , as a number ( rather than as a symbol for differential operations ), and expand it according to the binomial formula , we get_{}_{}
_{}= =_{}_{}
2 ^{0} _{}
_{}
_{}
In particular , when x _{0} =0, y _{0} =0 , we get
[ Maclaurin formula ]
f ( x , y )= _{} _{}
Similar formulas exist for multivariate functions of more than two variables .
[ Taylor series ] In the above Taylor formula 2° , if the expansion is carried out to any high power of ( ) _{}and ( ) _{}, there are
f ( x , y )= _{}
_{}
_{}
Regardless of whether it converges or not , and whether its sum equals f ( x , y ), it is called a Taylor series of f ( x , y ) .
[ Maclaurin series ] In the above Maclaurin formula , if the expansion is carried out to any high power of x , y , there is
f ( x , y )= f (0,0)+ _{}_{}
Whether it converges or not , and whether its sum equals f ( x , y ), it is called the Maclaurin series of f ( x , y ) .
Eight, power series
1 . single variable power series
[ definition ] a series of the following form
_{} ( 1 )
(where a _{0} , a _{1} , _{}are real constants) is called the power series of x . More generally, the series
_{}
( where a is a real constant ) is also called a power series .
[ Absolutely convergent ] If the series ( 1 ) converges when x = , then the series ( _{}1 ) converges absolutely for any value of x satisfying  x  <  ._{}
[ Convergence radius and convergence interval ] For any power series, there is a number R (0 ≤ R <+ ∞ ), so that when  x  < R , the series absolutely converges, when  x  > R , The series diverges . This number R is called the radius of convergence of a given series, the interval (R, R ) is called its convergence interval, and at the two endpoints of the interval x = R and x =  R , the series may Convergence can also diverge .
The radius of convergence R can be calculated according to the Cauchy  Hadamard formula
_{}
or the formula R = _{}
Compute ( if limit exists ).
[ Abel's theorem ] If the power series S ( x ) = (  x < R ) converges at the endpoint x = R of the convergence interval, then_{}
S ( R )=_{}
[ Internally closed uniform convergence ] If the radius of convergence of the series ( 1 ) is equal to R , then for any satisfying 0<< R , the series ( 1 ) converges uniformly on the interval [ , ] ._{}_{}_{}_{}
[ Continuous ] The sum of the power series is continuous at every point in the convergence interval .
[ itemwise integration ] At any point x within the convergence interval of the series ( 1 ) , there is
_{}
where S ( x ) represents the sum of series ( 1 ) .
[ itemwise differentiation ] The sum S ( x ) of a power series ( 1 ) is differentiable at any point within the convergence interval of this series .
_{}
has the same radius of convergence as ( 1 ), and the sum of this series is equal to ._{}
[ HigherOrder Derivative ] If the series ( 1 ) has a radius of convergence R , then its sum S ( x ) has an arbitrary derivative at any point in the interval ( , R ) , and the function _{}( n =1, ) is The sum of the series (whose radius of convergence is also R ) obtained n times of the term differential order ( 1 )_{}_{}
_{}= ( < x < R )_{} _{}
2 . Multivariate Power Series
[ Power series of two variables ] According to the positive integer powers of the variables x and y , the form is as follows
_{} ( 2 )
The heavy series is called the power series of bivariate x , y .
The study of the range of convergence of multivariate power series differs in many ways from univariate, but there are still
Theorem If the series ( 2 ) converges when x = x _{0} , y = y _{0 , then when} _{}_{}
 x < x _{0} , y < y _{0} 
, the series also converges .
[ Convergence range ] If M is a region of two variables x , y , where the power series ( 2 ) converges at each point, and diverges at the points outside it, it may be possible at the boundary points Divergence, it may also converge . Then the region M is called the convergence range of the power series ( 2 ) .
The convergence range of a bivariate power series is not necessarily of the form  x < R _{1} ,  y < R _{2} , e.g.
1 ° series
_{}
The range of convergence is  x <1 ,  y <1.
2° series
_{}
Condensed everywhere .
3° series
_{}=1+ x + + xy + x ^{2 }y + x ^{3 }y + + x ^{2 }y ^{2} +_{}^{}^{}_{}^{}^{}_{}
( =(1+ x + )[1+ xy + ]= _{}_{})_{}
The range of convergence is  x <1 ,  xy <1.
The above results are easy to generalize to multivariable power series .
3 . Power series expansion of a function
[ Uniqueness theorem of power series ] If the function f ( x ) ( or f ( x , y ) ) can be expanded into a power series at x = 0 ( or x = 0, y = 0)
f ( x )=_{}
or _{}
Then this power series is its Maclaurin series .
[ Existence Theorem of Power Series ]
1° if the function f ( x ) has any derivative at x = 0 and when ≤ x ≤ R _{}
_{}
where R _{n} ( x ) is the remainder of the Maclaurin formula, then the function f ( x ) can be expanded into a power series in the interval ≤ x ≤ R. In fact, it can be proved that there is a function generated by the function f ( x ) Maclaurin series, although it converges, its sum is not equal to f ( x )._{}
2° If the function f ( x , y ) has an arbitrary partial derivative at the point (0,0) and when ( x , y ) is a point on a region M on the xy plane
_{}
where R _{n} ( x , y ) is the remainder of the Maclaurin formula, then the function f ( x , y ) can be expanded into a power series on the region M.
The above theory can be easily extended to the case of multivariate functions with more than two variables .
Nine , the power series expansion table of functions on the real number field
function 
power series expansion 
Region of convergence 
[ Binomial ] _{} ( m >0) function 
_{} _{} ( When m is a positive integer, only m +1 items are included ) power series expansion 
_{}1 Region of convergence 
_{} _{} ( m >0) 
_{} _{} _{} _{} 
_{}1 _{}_{} 
_{} _{} 
_{} _{} 
_{}<1 
_{} 
_{} 
_{}<1 
_{} 
_{} 
_{}<1 
_{} 
_{} 
_{}<1 
_{} ( m >0) 
_{} _{} _{} 
_{}<_{} 
_{} 
_{} 
_{}<_{} 
_{} ( p > 0 or q > 0) 
_{} 
_{}≤ 1 
[ trigonometric functions ] _{} 
_{} 
_{}< ∞ 
_{} 
_{} 
_{}< ∞ 
_{} 
_{} 
_{}< ∞ 
function 
power series expansion 
Region of convergence 
_{} _{} 
_{} _{} _{} _{} ( where B _{n} is the Bernoulli number, the same below, see the attached table on page 231 ) 
_{}< ∞ _{}<_{} 
_{} 
_{} _{} 
0 <<_{}_{} 
_{} 
_{} _{} ( where E _{n} is the Euler number, see the attached table on page 231 ) 
_{}<_{} 
_{} 
_{} _{} 
0 <<_{}_{} 
[ inverse trigonometric function ] _{} 
_{} _{} 
_{}< 1 
_{} 
_{} _{} 
_{}< 1 
_{} 
_{} 
_{}< 1 _{}_{}1 
function 
power series expansion 
Region of convergence 
_{} [ exponential function ] _{} 
_{} _{} 
_{}< 1 _{}< ∞ 
_{} 
_{} 
_{}< ∞ 
_{} 
_{} _{} 
_{}<_{} 
_{} 
_{} 
_{}< ∞ 
_{} 
_{} 
_{}< ∞ 
_{} 
_{} 
_{}<_{} 
[ logarithmic function ] _{} 
_{} 
x >0 
_{} 
_{} 
0<_{} 
_{} 
_{} 
x >_{} 
_{} 
_{} 
_{}1 
_{} 
_{} 
_{} 
_{} ( a > 0) 
_{} 
_{} 
_{} 
_{} 
_{}<1 
function 
power series expansion 
Region of convergence 
_{} 
_{} 
_{}>1 
_{} 
_{} _{} 
_{}>1 
_{} 
_{} 
0 <<_{}_{} 
_{} 
_{} _{} 
_{}<_{} 
_{} 
_{} _{} 
0 <<_{}_{} 
[ Hyperbolic function ] sh x 
_{} 
_{}<_{} 
ch x 
_{} 
_{}<_{} 
th x 
_{} _{} 
_{}<_{} 
cth x 
_{} _{} 
0 <<_{}_{} 
sech x 
_{} _{} 
_{}<_{} 



function 
power series expansion 
Region of convergence 
csch x [ inverse hyperbolic function ] Arsh x = _{} 
_{} _{} _{} _{} 
0 <<_{}_{} _{}<1 
Arsh x 
_{} _{} 
_{}>1 
Arch x ( dual value ) Arth x = _{} Arcth x = _{} 
_{} _{} _{} _{} _{} 
_{}>1 _{}<1 _{}>1 
Those who bid * in the table should keep in mind .
Attachment: Bernoulli number B _{n} and Euler number E _{n} table
n 
B _{n} 
En _{_} 
1 
_{} 
1 
2 
_{} 
5 
3 
_{} 
61 
4 
_{} 
1 385 
5 
_{} 
50 521 
n 
B _{n} 
En _{_} 
6 
_{} 
2 702 765 
7 
_{} 
199 360 981 
8 
_{} 
19 391 512 145 
9 
_{} 
2 404 879 675 441 
10 
_{} 
370 371 188 237 525 
_{}
_{}
* See Chapter 21 for the concept of neighborhood, where the field of M _{0} refers toa rectangle containing M _{0} _{}_{}