y0(3)

💖 NAME 💖

y0, y0f, y0l, y1, y1f, y1l, yn, ynf, ynl — Bessel functions of the second kind, bestie. yes we're doing special functions today, get comfy 🫶

📚 LIBRARY 📚

Math library (libm, -lm). so like. you will be linking -lm. it's giving "forgot the flag, spent an hour confused" energy, don't be that girl 💅

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✨ SYNOPSIS ✨

#include <math.h>

double y0(double x);
double y1(double x);
double yn(int n, double x);

float y0f(float x);
float y1f(float x);
float ynf(int n, float x);

long double y0l(long double x);
long double y1l(long double x);
long double ynl(int n, long double x);

🧚 Feature Test Macro Requirements for glibc (see feature_test_macros(7)):

y0(), y1(), yn():
    _XOPEN_SOURCE
        || /* Since glibc 2.19: */ _DEFAULT_SOURCE
        || /* glibc <= 2.19: */ _SVID_SOURCE || _BSD_SOURCE

y0f(), y0l(), y1f(), y1l(), ynf(), ynl():
    _XOPEN_SOURCE >= 600
        || (_ISOC99_SOURCE && _XOPEN_SOURCE)
        || /* Since glibc 2.19: */ _DEFAULT_SOURCE
        || /* glibc <= 2.19: */ _SVID_SOURCE || _BSD_SOURCE

okay so the /* Since glibc 2.19: */ and /* glibc <= 2.19: */ bits are load-bearing — they tell you which glibc era each alternative applies to. do not delete them, they're not decoration 🎀

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🌸 DESCRIPTION 🌸

so here's the tea ☕:

• y0() and y1() hand you back Bessel functions of x of the second kind, of orders 0 and 1 respectively.
• yn() hands you back the Bessel function of x of the second kind of order n.

The value of x must be positive. that's not a suggestion, that's a requirement, periodt ✋

The y0f(), y1f(), and ynf() gals are the versions that take and return float values. The y0l(), y1l(), and ynl() versions take and return long double values. same vibes, bigger/smaller pants 💜

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🔮 RETURN VALUE 🔮

On success, these functions return the appropriate Bessel value of the second kind for x. slay ✨

• If x is a NaN, a NaN is returned. (garbage in, garbage out, iconic behaviour)
• If x is negative, a domain error occurs, and the functions return -HUGE_VAL, -HUGE_VALF, or -HUGE_VALL, respectively. (POSIX.1-2001 also allows a NaN return for this case.)
• If x is 0.0, a pole error occurs, and the functions return -HUGE_VAL, -HUGE_VALF, or -HUGE_VALL, respectively. 💀
• If the result underflows, a range error occurs, and the functions return 0.0
• If the result overflows, a range error occurs, and the functions return -HUGE_VAL, -HUGE_VALF, or -HUGE_VALL, respectively. (POSIX.1-2001 also allows a 0.0 return for this case.)

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🚨 ERRORS 🚨

See math_error(7) for information on how to determine whether an error has occurred when calling these functions. iykyk.

The following errors can occur:

• Domain error: x is negative — errno is set to EDOM. An invalid floating-point exception (FE_INVALID) is raised.
• Pole error: x is 0.0 — errno is set to ERANGE and an FE_DIVBYZERO exception is raised (but see BUGS). 👀
• Range error: result underflow — errno is set to ERANGE. No FE_UNDERFLOW exception is returned by fetestexcept(3) for this case.
• Range error: result overflow — errno is set to ERANGE (but see BUGS). An overflow floating-point exception (FE_OVERFLOW) is raised.

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👑 ATTRIBUTES 👑

For an explanation of the terms used in this section, see attributes(7). (the girls that know, know)

• y0(), y0f(), y0l() — Attribute: Thread safety — Value: MT-Safe
• y1(), y1f(), y1l() — Attribute: Thread safety — Value: MT-Safe
• yn(), ynf(), ynl() — Attribute: Thread safety — Value: MT-Safe

all of them MT-Safe, every single one, no exceptions, main character energy 🌈

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📜 STANDARDS 📜

• y0()
• y1()
• yn() — POSIX.1-2008.

Others: BSD.

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🦋 HISTORY 🦋

• y0()
• y1()
• yn() — SVr4, 4.3BSD, POSIX.1-2001.

Others: BSD.

(the Others being the f and l variants, and yes they're BSD-flavoured 🍓)

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💀 BUGS 💀

bestie sit down, we gotta talk about the old days:

• Before glibc 2.19, these functions misdiagnosed pole errors: errno was set to EDOM, instead of ERANGE and no FE_DIVBYZERO exception was raised. not the vibe 😭
• Before glibc 2.17, did not set errno for "range error: result underflow".
• In glibc 2.3.2 and earlier, these functions do not raise an invalid floating-point exception (FE_INVALID) when a domain error occurs.

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🫶 SEE ALSO 🫶

• j0(3)

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and that's the whole page, angel 💖 this one's from Linux man-pages 6.20, dated 2026-02-08 — go forth and Bessel responsibly, ily, bye 🎀✨