Differential and Integral Calculus
by R.Courant
Summary of Important Theorems and Formulae
2. Convergence of Sequences and Series
2.1 Infinite sequences 1.6:
Cauchy's convergence test 1.6.2. A sequence of numbers an is convergent if, and only if, there exists for every positive constant e a number N such that
Operating with
limits 1.6.4. If 
exist, then
2.2 Infinite Series 8:
Cauchy's convergence test 8.1.1 The series San converges if, and only if, for there exists for every positive quantity e a number N such that
Note: All the following criteria are sufficient, but not necessary.
Principal of Comparison of Series 8.2 San converges if there exist numbers bn such that bn ³ |an| for all values of n and Sbn converges.
Ratio test and and root test 8.2.2 San converges if there is a number N, and also a number q < 1 such that
for all values n > N; in particular, if there is a number k < 1 such that
Leibnitz's Test 8.2 San converges if the terms have alternating signs and |an| tends monotonically to zero.
3.1 General rule (Fundamental Ideas 2.3:
Chain Rule
with corresponding formulae for uxy and uyy 10.4.2
Implicit Functions
Functions expressed in terms of a parameter
Inverse functions
3.2 Special Formulae 2.3.3 3.1.3 3.3.3 3.6 3.8.3
4.1 General Rules (Fundamental Ideas)
Estimation of Integrals
Integration by Parts
Method of Substitution
Link between Differentiation and Integration
Improper Integrals 4.8
If f(x) is
continuous except at the point x = b, where it
becomes infinite, 
is (absolutely) convergent, if in the
neighbourhood of x = b
where n > 1, for values of x ³ A. 4.8.3
4.2 Special Formulae 2.2 2.7.2 3.2.2 3.3.4 3.6 4.1 4.2.1 4.2.3 4.4.2
Recurrence Relations 4.4.3
4.3 Integration of Special Functions
4.3.1 Rational functions These are reduced to the following three fundamental types by resolution into partial fractions 4.5.1
the integral on the right hand side being evaluated by the last recurrence relation above;
where the integral on the right hand side is of the preceding type.
In the sequel, R denotes a rational function.
4.3.2
4.6.3
Substitution: t = tan x/2, so that
However, if R is an even function or only involves tan x, the following substitution is more convenient:
4.3.3 
4.6.3
Substitution t = tanh x/2, so that
4.3.4 
Substitution t = emx, dx/dt = 1/mt.
4.3.5 
4.6.4
Substitution:
4.3.6 
4.6.5
Substitution:
4.3.7 
4.6.6
Substitution:
4.3.8 
4.6.7
The substitution 
reduces this integral to one of the
preceding three types.
4.3.9 
4.6.8
Substitution: 
4.3.10 
4.6.8
Substitution:
5. Uniform Convergence and Interchange of Infinite operations
For the definition of uniform convergence go to 8.4.2.
A series, which is uniformly convergent in a closed interval and the terms of which are continuous functions, represents a continuous function in the interval 8.4.3.
If |f(x)| £ an and San converges, Sfn(x) converges uniformly ( and absolutely). 8.4.2.
Interchange of summation and differentiation 8.4.5
Any convergent series of continuous functions may be differentiated term by term, provided the resulting series converges uniformly.
Interchange of summation and integration 8.4.4
Any uniformly convergent series of continuous functions may be integrated term by term. The resulting series also converges uniformly.
Stirling's Formula
Wallis' Product 4.4.4 A7 9.4.7
Infinite products A8.3
Definition of the Gamma function 4.8.4
if x is a positive integer n,
Order of magnitude of functions 3.9
Orthogonality relations of the trigonometric functions 4.3
8. Mean Value theorems
Mean value theorem of the differential calculus 2.3.8
If f(x) = f(x + h) = 0, this yields Rolle's theorem 2.3.8: Between two zeros of the function lies always a zero of the derivative.
Generalized Mean Value theorem A2.2 A3.3.3
where x is a value between a and b.
Taylor's theorem 6.2
with the remainder
Mean value theorem of the integral calculus 2.7.1
9. Expansion in Series: Taylor Series, Fourier Series
1. Power series
Definition 8.5
9.1.1 Power series in general
Any power series
in one variable has a radius of convergence r (which may be zero or infinite); the series converges when |x| < r; in fact, it converges uniformly and absolutely in every interval |x| £ h, where h < r; when |x| > r; the series diverges 8.5.1
If the remainder in Taylor's theorem tends to zero as n increases, we have the infinite power series 6.2.3
9.1.2 Special Taylor series 6.1.1 6.3 8.6 A8.4
where the Bn are Bernoulli numbers A8.4.
9.1.3 Binomial series
9.1.4 Elliptic integral
9.2 Fourier series
If the function f(x) is sectionally smooth in the interval -p £ x £ p, i.e., if its first derivative is sectionally continuous, the Fourier series
is absolutely convergent throughout the entire interval. If f(x) has a finite number of jump discontinuities, while f '(x) is elsewhere sectionally continuous, the series converges uniformly in every closed subinterval which contains no discontinuities of f(x). At every point at which f(x) is continuous, the series represents the value of the function f(x), while at every point of discontinuity of f(x) it represents the arithmetic mean of the right hand and left hand limits of f(x) 9.5.
The following rule holds only for maxima and minima in the interior of the region under consideration.
In order that x may be an extreme value of the function y = f(x), f '(x) must vanish. When this condition is satisfied, there is a maximum or minimum, if the first non-vanishing derivative of f(x) is of even order; if it is of odd order, there is neither a maximum nor a minimum. In the former case, there is a maximum or a minimum according to whether the sign of the first non-zero derivative is negative or positive 3.5.
In what follows, x, h are current co-ordinates.
Equation of the curve:
Equation of the tangent at the point (x,y) 5.1.3
Equation of the normal at the point (x,y) 5.1.3
Curvature 5.2.6
Radius of curvature 5.2.6
Evolute (locus of centre of curvature) 5.2.6 A5.1
Involute A5.1
where a is an arbitrary constant and s is the length of arc measured from a given point.
Point of inflection 3.5.1 5.1.5
Necessary condition for a point of inflection is
Angle between two curves 5.1.3
In particular, the curves are orthogonal, if
the curves touch, if
Two curves y = f(x), y = g(x) have contact of order n at a point x, if 6.4
12. Length of Arc, Area, Volume
Length of arc 5.2.5
Let a plane curve be given by the equations
The length of arc is
Area of plane surface
The area bounded by the curve
and two radius vectors q0, q1, where r, q are polar co-ordinates, is given by 5.2.4
The area, enclosed by the curve
the two ordinates x = x0, x = x1 and the x-axis, is 2.1.2
Volume
The volume lying over a region R and bounded above by the surface with the equation
is given by 10.6
End
