# Book:Murray R. Spiegel/Mathematical Handbook of Formulas and Tables/Chapter 35

## Murray R. Spiegel: Mathematical Handbook of Formulas and Tables: Chapter 35

Published $\text {1968}$

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## $35 \quad$ Miscellaneous Special Functions

### Error Function $\ds \map \erf x = \frac 2 {\sqrt \pi} \int_0^x e^{-u^2} \rd u$

$35.1$: Power Series Expansion for Error Function
$35.2$: Asymptotic Expansion for Error Function
$35.3$: Properties of Error Function
$35.3.1$: Error Function is Odd
$35.3.2$: Error Function of Zero
$35.3.3$: Limit to Infinity of Error Function

### Complementary Error Function $\map \erfc x = 1 - \map \erf x$

$35.4$: Power Series Expansion for Complementary Error Function
$35.5$: Asymptotic Expansion for Complementary Error Function
$35.6$: Properties of Complementary Error Function
$35.6.1$: Complementary Error Function of Zero
$35.6.2$: Limit to Infinity of Complementary Error Function

### Exponential Integral $\ds \map \Ei x = \int_x^\infty \frac {e^{-t} } t \rd t$

$35.7$: Characterization of Exponential Integral Function
$35.8$: Power Series Expansion for Exponential Integral Function
$35.9$: Asymptotic Expansion for Exponential Integral Function
$35.10$: Limit to Infinity of Exponential Integral Function

### Sine Integral $\ds \map \Si x = \int_0^x \frac {\sin u} u \rd u$

$35.11$: Power Series Expansion for Sine Integral Function
$35.12$: Asymptotic Expansion for Sine Integral Function
$35.13$: Properties of Sine Integral Function
$35.13.1$: Sine Integral Function is Odd
$35.13.2$: Sine Integral Function of Zero
$35.13.3$: Limit to Infinity of Sine Integral Function

### Cosine Integral $\ds \map \Ci x = \int_x^\infty \frac {\cos u} u \rd u$

$35.14$: Characterization of Cosine Integral Function
$35.15$: Power Series Expansion for Cosine Integral Function plus Logarithm
$35.16$: Asymptotic Expansion for Cosine Integral Function
$35.17$: Limit to Infinity of Cosine Integral Function

### Fresnel Sine Integral $\ds \map {\mathrm S} x = \sqrt {\frac 2 \pi} \int_0^x \sin u^2 \rd u$

$35.18$: Power Series Expansion for Fresnel Sine Integral Function
$35.19$: Asymptotic Expansion for Fresnel Sine Integral Function
$35.20$: Properties of Fresnel Sine Integral Function
$35.20.1$: Fresnel Sine Integral Function is Odd
$35.20.2$: Fresnel Sine Integral Function of Zero
$35.20.3$: Limit to Infinity of Fresnel Sine Integral Function

### Fresnel Cosine Integral $\ds \map {\mathrm C} x = \sqrt {\frac 2 \pi} \int_0^x \cos u^2 \rd u$

$35.21$: Power Series Expansion for Fresnel Cosine Integral Function
$35.22$: Asymptotic Expansion for Fresnel Cosine Integral Function
$35.23$: Properties of Fresnel Cosine Integral Function
$35.23.1$: Fresnel Cosine Integral Function is Odd
$35.23.2$: Fresnel Cosine Integral Function of Zero
$35.23.3$: Limit to Infinity of Fresnel Cosine Integral Function

### Riemann Zeta Function $\ds \map \zeta x = \frac 1 {1^x} + \frac 1 {2^x} + \frac 1 {3^x} + \ldots$

$35.24$: Integral Representation of Riemann Zeta Function in terms of Gamma Function
$35.25$: Functional Equation for Riemann Zeta Function
$35.26$: Riemann Zeta Function at Even Integers

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