An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

viii Contents.
Page
series. 47. Logarithmic series. 48. Circular series. 49. Even and odd
functions. 50. The general problem of the developability of a function
in a series of powers.
Ninth Chapter: Functions of more than one independent variable. 86
~ 51. The function of two variables and its geometric exposition.
52. Continuity. Examples. Uniform continuity in the domain. 53. First
partial differential quotients. The total differential. 54. Higher partial
derived functions. Interchange of the order in differentiating. 55. Higher
total differentials. 56. Taylor's expansion.
Tenth Chapter: Implicit functions. Application of Taylor's
series to evaluate quotients apparently indeterminate. 101
~ 57. The implicit function. Its differential quotient determinable.
58. The implicit algebraic function. 59. Continuation from a point.
60. Differential quotient at a multiple point requires the determination of a
quotient of the form o. 61. Differential quotient at infinitely distant
points. 62. Determination of a quotient of the form.
Second Book. (Pp. 110 —168.)
Complex numbers and functions of complex numbers.
First Chapter: The complex number and the operations of
arithmetic......................... 110
~ 63. The imaginary unit. 64. Real and imaginary differ in conception
not in application. 65. The complex number. 66. Geometric interpretation.
67. Representation of the complex by modulus and amplitude. 68. Complex
numbers form a group complete in themselves. 69. Summation. 70.
Multiplication. 71. Division. 72. Power with a real exponent. 73. Power
with a complex exponent. 74. Logarithm. 75, Power with a complex
base and a complex exponent.
Second Chapter: Complex series. Complex variable. Functions
of a complex variable.................. 121
~ 76. Complex series. 77. Absolute convergence. 78. Multiplication
of two series. 79. Complex variable. Geometric representation of infinite
values. 80. Function. 81. Continuity of a function. 82. Examples:
I. The integer rational, II. the fractional rational function. III. Explicit
irrational function. Branching points. IV. Exponential function. The
essential singular point. V. Logarithm. 83. Any convergent complex
series of powers is one-valued and continuous. 81. The differential
quotient. Analytic functions. 85. Transformation by analytic functions.
86. Examples. The complex series of powers is an analytic function.
Third Chapter: The vanishing values of a series of powers,
specially those of the integer rational algebraic
function.................. 149
~ 87. Finite number of zero points in a finite domain. 88. Separation
of factors. 89. Cauchy's theorem. 90. The fundamental theorem of
algebra.
Fourth Chapter: The implicit algebraic function......... 154
~ 91. The problem. 92. Non-essential singular and critical points.
93. Continuity of the algebraic function. 94. Its analytic property.
95. Its uniqueness along definite paths. 96. Riemann's n-leaved surface.
97. The point infinity. 98. Examples. 99. The problem of calculation.
Third Book. (Pp. 169-320.)
Integrals of functions of real variables.
First Chapter: The definite and the indefinite integral..... 169
~ 100. The theorem of the mean value holds for continuous functions
whose progressive differential quotient is continuous. 101. The problem