<p dir="auto">My issue is about scipy.special.owens_t not returning the correct value when the second argument is infinity.</p>
<p dir="auto">For example, scipy.special.owens_t(0,numpy.inf) should return 0.25, not 0.5.</p>
<p dir="auto">It looks as if scipy.special.owens_t(x,numpy.inf) always returns twice the value that it should do. Please see <a href="https://en.wikipedia.org/wiki/Owen%27s_T_function" rel="nofollow">https://en.wikipedia.org/wiki/Owen%27s_T_function</a> for some basic information about the function. I have given some example code where the second argument gets successively larger, and it is clear what the asymptotic value should be.</p>
<h4 dir="auto">Reproducing code example:</h4>

<div class="snippet-clipboard-content notranslate position-relative overflow-auto" data-snippet-clipboard-copy-content="large = 1e16
larger = 1e32
largest = numpy.inf
scipy.special.owens_t(0, large) # almost equal to 0.25
scipy.special.owens_t(0, larger) # equal to 0.25
scipy.special.owens_t(0, largest) # equal to 0.5 It should be 0.25
"><pre class="notranslate"><code class="notranslate">large = 1e16
larger = 1e32
largest = numpy.inf
scipy.special.owens_t(0, large) # almost equal to 0.25
scipy.special.owens_t(0, larger) # equal to 0.25
scipy.special.owens_t(0, largest) # equal to 0.5 It should be 0.25

</code></pre></div>
<h4 dir="auto">Scipy/Numpy/Python version information:</h4>

<div class="snippet-clipboard-content notranslate position-relative overflow-auto" data-snippet-clipboard-copy-content="1.5.2 1.19.1 sys.version_info(major=3, minor=8, micro=5, releaselevel='final', serial=0)"><pre class="notranslate"><code class="notranslate">1.5.2 1.19.1 sys.version_info(major=3, minor=8, micro=5, releaselevel='final', serial=0)
</code></pre></div>
<h3 dir="auto">Describe your issue.</h3>
<p dir="auto">Hello,</p>
<p dir="auto"><code class="notranslate">owens_t(0, inf)</code> gives <code class="notranslate">0.5</code>. However <code class="notranslate">owens_t(a, inf)</code> is <code class="notranslate">atan(a)/(2*pi)</code> , so  <code class="notranslate">owens_t(0, inf)</code> shoud give <code class="notranslate">0.25</code>.</p>
<h3 dir="auto">Reproducing Code Example</h3>
<div class="highlight highlight-source-python notranslate position-relative overflow-auto" dir="auto" data-snippet-clipboard-copy-content="from scipy import special
from math import atan, pi, inf

special.owens_t(0,inf)
# 0.5
atan(inf)/2/pi
# 0.25"><pre class="notranslate"><span class="pl-k">from</span> <span class="pl-s1">scipy</span> <span class="pl-k">import</span> <span class="pl-s1">special</span>
<span class="pl-k">from</span> <span class="pl-s1">math</span> <span class="pl-k">import</span> <span class="pl-s1">atan</span>, <span class="pl-s1">pi</span>, <span class="pl-s1">inf</span>

<span class="pl-s1">special</span>.<span class="pl-en">owens_t</span>(<span class="pl-c1">0</span>,<span class="pl-s1">inf</span>)
<span class="pl-c"># 0.5</span>
<span class="pl-en">atan</span>(<span class="pl-s1">inf</span>)<span class="pl-c1">/</span><span class="pl-c1">2</span><span class="pl-c1">/</span><span class="pl-s1">pi</span>
<span class="pl-c"># 0.25</span></pre></div>
<h3 dir="auto">Error message</h3>
<div class="highlight highlight-source-shell notranslate position-relative overflow-auto" dir="auto" data-snippet-clipboard-copy-content="No error message"><pre class="notranslate">No error message</pre></div>
<h3 dir="auto">SciPy/NumPy/Python version information</h3>
<p dir="auto">scipy 1.7.1, python 3.9.5</p>